Rating: Impossible (requires cheating)
There is no reasonable procedure that allows you to win this scenario. One way or another, you are going to have to cheat. Whether you prefer to memorize a very, very long series of hand motions, or save the game every 15 seconds, or use a cheat code directly, it's really all the same. This is one of those scenarios where you have to master the pound-click (pound the left mouse button to ensure that you actually select something immediately) and the machine-gun right click (right-click several times rapidly to ensure that the selected unit actually moves immediately).
The map is crawling with enemy troops, and their quality is generally better than yours. The only area where you are superior is in heavy artillery. The enemy also has an economy, so if you move slowly, you will lose in any case. You do not have time to explore; you need to know where everything is. It sort of makes the "fog of war" feature pointless. You cannot restore any damage at the beginning, and you will never be able to. You get no reinforcements at all. This means that you pretty much need to avoid all damage, and the interface is too clumsy for that to be possible, even if you move your hands infinitely fast.
There are a couple of techniques that you have to use. First, keep all of your units together. You need to maximize the fire on each target. Second, keep your siege weapons right beside a Transport when fighting enemy warships. They need to jump into the Transport the instant they come under fire. Third, units that come under fire need to take evasive action while other units shoot. Fourth, you need to keep your units moving all the time, especially the ships.
Save the game frequently. Unfortunately, AoE save files often get corrupted, especially if you save to the same file multiple times. It might help to use multiple save files. If it gets too frustrating, just enter the "winner" cheat code.
Your first objective is to take out the Izumo production. They have 17 Houses, and can support far more than 50 units. You can see from Achievements/Timeline that they have 33 units at the start of the scenario, and that they are definitely growing. Taking out their Houses won't really help, unless you get a lot of them. Taking out their dock is a possibility, but that might be too costly until later. Taking out their villagers, who are collecting wood, is easiest, assuming that you know where they are. Unfortunately, Izumo may have a wood stockpile, and you can't do anything about that.
Island 0: This is your starting position. You have 7 different types of units. Assign a control code to each type. You should probably load the artillery onto the small Transport, and everyone else onto the large Transport. Head west and unload the artillery on the first island you encounter.
Island 1: This is a large island. Move your catapult far enough inland to reveal the whole island. You will probably engage an enemy warship as it passes the south side of this island. Load up again, move around to the south of island 1, and land the artillery on the island immediately to the south of it.
Island 2: This is a large island. There is some hostile wildlife here. Nuke the nearest elephant with your catapult, and the nearest lion with your Ballista. There is an Artifact here, which you control as soon as you land. Use it to explore this island. Load your artillery up again, move around to the west of island 1, and land it there. You will probably engage an enemy warship around this time.
Island 3: This island is NW of island 1. The side facing you is all cliff. There are a bunch of enemy units on top of the cliff. Pick off the two archers with your catapult. If you shoot at the archers directly, they will just run away; shoot at the units beside them and kill the archers with area effect damage. Load up again, move directly south, and land the artillery on the island west of island 2.
Island 4: This is a large island. There are two archers here. Pick them off with your artillery. Keep in mind that it is hard to get hits with indirect fire on hilly terrain; you generally have to shoot past the target. Take the shots, but be prepared to leap back into the Transport at a moment's notice. Make sure that you don't take any damage here. There is an Artifact on this island, but it doesn't come over to you immediately, so ignore it for now. Load up again and head SE, around the south coast of island 2.
Island 2 again: Land your artillery again briefly. Pick off an elephant or a lion if they are in range. Load up again, and head down along the SE map edge. Take the Artifact with you this time, because you don't know if you will return to this island.
Island 5: This island, in the south corner of the map, is heavily forested. Izumo has a woodcutting operation on the south tip of this island. Kill the villagers. If they run away, blow up the Storage Pit. Hopefully, this will stop the Izumo warship production. Actually, you know that Izumo had three workers at the very beginning of the scenario and produced three more. The workers on this island must be the initial ones, because they could not have been produced. Load up your artillery again and land it on the island to the west, just across the channel.
Island 6: This long, thin island runs along the SW map edge. Izumo has a bunch of Houses here. Leave them alone for now. Load your artillery up again and land it on the south coast of island 4.
Island 4 again: This island is now clear and is quite safe for your artillery to operate from, because it is large enough that Izumo warships can't reach the middle. You would like to camp out on this island and blow up all the Izumo warships with indirect fire using your superior range, but they probably won't oblige you totally. Therefore you will need to use Shoot-and-Skedaddle tactics. Be careful! Your catapult doesn't skedaddle very quickly, and will probably take some hits in the process. However, in conjunction with your War Galleys and Ballista, which leaps out of its Transport to take a shot and then leaps right back into the Transport, you should be able to destroy a bunch of Izumo warships without taking too much damage. When it appears that no more Izumo warships are in the area, load up your artillery again and land it on the south end of island 6. You will have taken over the Artifact on island 4 in passing. You can leave it there for now.
Island 6 again: Advance slowly along the map edge, blowing up all the Houses as you go. There are 12 of them, and the loss of that many Houses should restrict the Izumo production. Stay alert in case any Izumo warships show up, because your artillery will be far from its Transport. Advance your War Galleys along the channel to protect the flank. There is an Izumo dock on the north tip of this island. Blow it up, and the Izumo production is permanently finished. As your warships advance out of the channel, they locate another island to the NW. You should see some Izumo fishing ships in the area, just standing around, now that their dock is gone. These are the new-production Izumo workers.
Island 7: This is a small island NW of island 6. There is an Artifact here. It comes over to you immediately. Because of this, and the alligator, you know that there are no Izumo units nearby. Land your artillery here briefly and collect the Artifact. There are shallows leading away from this island to the NW.
At this point, you should stop and take stock. Your infantry should be in perfect health. Your Transports should be in excellent condition. Your artillery should be in excellent condition overall. At least one of your War Galleys should be in excellent condition. You have done grievous damage to the Izumo economy and production, and perhaps destroyed them utterly. You have explored over half the map, and there just aren't a lot of places the Izumo might still have infrastructure. Most likely, all you have to do now is hunt down and kill all the Izumo units guarding the remaining Artifacts, and all the advantages are now with you. Your primary advantage is now the huge range of your catapult, and you should be able to use it to kill all the enemy units. Just creep your catapult forward and eliminate any opposition with indirect fire before they know what hit them. You still need to be prepared to withdraw at a moment's notice, however.
It doesn't really matter exactly how you proceed after this, because the Izumo are now static and can only wait to be killed. However, there is an optimum path, and the general idea is that you want to do the safe stuff before you take any risks at all.
The Archipelago: There is an archipelago in the west corner of the map - a bunch of small islands, all connected by shallows. There are some Izumo archers here, and one Artifact. Clear this area next, then move towards the south end of island 3.
Island 3 again: There are a bunch of Izumo troops here. Slay them all, and collect the fifth Artifact. Deposit them all somewhere convenient, or carry them along with you; it doesn't matter. Move your catapult along the north coast of this island. Then move to island 1 again, and move along its north coast. The channel between islands 1 and 3 is too narrow for Transports and warships. Move to the north coast of island 2.
Island 2 again: Finish exploring this island, killing off any predators as necessary. Watch out for those Lion Kings! When they hide in a forest, they can be very hard to see, and yes, they will eat catapults.
Island 8: This is a tiny island to the NE of island 2. There are two priests and a hero archer here. Slay them all. Ship your catapult to this island.
Island 9: This is a small island in the east corner of the map. The part facing you is all cliff. You can pick off a Guard Tower and a House from island 8.
To finish this phase of the scenario, return your catapult to island 0 and explore as much of the sea as you can. You will now have explored about 90% of the map, everything except the east and north map corners. The Izumo should be down to 7 units, of which 3 are fishing ships. You now have to take some risks, because you have to explore with your ships, which have a fairly short sight range, and running into some priests could ruin your day. Save the game at this point.
There are good procedures for minimizing the upcoming risks. Explore with your weaker War Galley alone. If it gets converted, you should be able to kill it off. Be prepared to evade ranged fire. Have your other units standing by for contingencies, close but out of range.
There is a small Izumo catapult on island 9. You can kill it fairly easily with your warships, but there isn't much point. There are also several Izumo Houses here. Cliffs prevent you from landing on this island.
There is a small island in the north corner of the map. There are three Izumo archers on it. They don't have Ballistics, so you should be able to get your War Galley in quite close to shore without taking any damage, as long as you have mastered arrow evasion tactics. Once you have located all three remaining Izumo units, withdraw and prepare your final assault.
Assuming that your large Transport is still in excellent condition, it should lead the assault. It can't be sunk before it debarks your 5 infantry right in the teeth of the enemy. Your weaker War Galley should go in last. The one thing you don't want to do is bring your catapult along in this attack. It will merrily blow up your own units. Collect the final Artifact, and that's all, folks.
This map shows the route you should follow up until the final assault on the main Izumo base. Troop movements are indicated with solid red arrows. Significant artillery attacks are indicated with dashed red arrows. Islands 0-9 are numbered, and artifacts are indicated with a red "A".