This is not actually a walkthrough, just some helpful info.
As the briefing would imply, this is mainly a water based map. Naturally the scenario is won or lost over how much naval control either side has.
The map is quite realistic and includes the southern part of Korea and the western part of Japan. So a look on a real map will help you to orientate yourself.
Stone is near useless on this map, since you can't build castles and towers won't be enough against cannon galleons(though they can help with the other ships in a pinch). Given that most of your offensive is going to be with ships, focus on getting a lot of wood first. Gold and food come next, though after researching most of the ship upgrades you can forget about food too and focus solely on wood and gold.
The towers around the wonder will help fend off Japan's initial fleet, but you should collect all your ships and wipe them out, including the ships that are waiting just outside the visible area off the eastern coast. Use hit and run tactics to lure them near your towers. Try not to lose too many of your ships, and remember to target cannon galleons as a priority. When you are done, you don't need to care for that coast anymore, the attacks will concentrate on the area where your two harbors are and the south coast (on a real map that's roughly the coast between Suncheon and Pohang).
Keep your ships healthy by repairing them (all of them are more advanced than what you can build initially), especially keep an eye on your two Cannon Galleons. Building a few fire ships may be a good idea. When you have defeated the initial batch of enemy ships, you will have a short pause before the next attacks, here you can and should start razing harbours already, but watch out for enemy towers and castles. After about 10 minutes, the Japanese will start sending transport ships with samurais, archers and occasionally monks to attack inland. None of these are a big problem to a town center garrisoned with atleast 10 villagers though, so ringing the town bell should be enough to dispose of them, with the towers helping.
Meanwhile you can take care of the Japanese Raiders in the north. Develop Chemistry, so you can build a cannon, and add a few knights to defend the cannon, this small army should be enough to defeat them. They only have a castle, a siege shop and three towers, two of which you don't need to take care of. Follow the road to the north across the bridge. When you are done, retreat to your town or delete the rest of the units, if you run into the population limit already, those land units are relatively useless from now on.
You will also find a transport ship there, use it to bring one knight on the other side of the river and ride towards the castles, actually a bit further than the flag, down to the beach, there you will find the turtle ships (two elite ones and Yi Sun Shin's ship that will heal itself over time and is quite strong), which are a very welcome reinforcement. This happens 8 minutes into the game (when the objective "Find the admiral" pops up), before that the gate will be locked, but it is advisable not to lose time, so to have the knight wait in front of the gate is not a bad idea.
Once you acquire these ships, the scenario becomes much easier, for not only are they pretty formidable ships, you also get to build more of them. Don't send them in to blow up the docks though, their cannons lack range and you don't want to lose them to towers/castles. Also be wary of any demolition ships the Japanese will send, as they tend to target turtle ships and deal a fair bit of damage to them.The Japanese Navy will have no villagers and no town center, so they will not rebuild anything you destroy. Just raze the harbors and you are pretty much done. Be aware that two harbors are a bit hidden in bays, in the western end of Japan, you have to sail around completely to find another harbor (on a real map that would be the Ariake bay), and another is behind a ford a bit further east (in real life that place would be around the town of Ube).
Also you can send a transport ship towards the west to the Chinese island to collect the Elite Chu Ko Nu and the Cannons, these will help in razing Japan completely, but actually you can do without. Not only does using them effectively need upgraded transport ships, but they also raise your population count, meaning that you get less room for ships which are handier here. As such, your regular fleet of cannon galleons supported by galleons, turtle ships and probably fast fire ships should raze the Japanese naval production just fine.
As you will be fighting on several fronts simultaneously, use the pause button often and micromanage all your units properly. Remember to focus on unlocking all the important naval upgrades, and keep pumping out as many ships as possible.