You start this scenario with an air of overconfidence. Your force composition suggests that you are about to engage in a final naval assault. Nothing could be further from the truth. In fact, your enemies are about to sweep you from the seas. At least, they will try to.
A force composition of 5 Triremes and 7 Catapult Triremes is pretty ridiculous. 10 Triremes and two Catapult Triremes would make sense. Catapult Triremes perform very poorly in naval combat. They are too slow and clumsy to get out of bad positions, and too small to stand up long, but their biggest deficiency is that they tend to blow up friendly ships. You don't ever want to have more than a few, because you have to micromanage them continuously, and that severely restricts the useful number. With Catapult Triremes, the extra ones are not just spares, they are liabilities. They are specialist ships, and you only want to have them to deal with certain specialized situations.
Normally, you want to keep your Catapult Triremes in protected locations. At the start of this scenario, there are no protected locations. The best that you can do is to shove them as far back in your cove as possible, and you actually need to do this the instant the scenario starts, otherwise at least one of them will be sunk before you can respond.
Your catapults actually make pretty good scouts, so put them to work on exploring. Move one towards your harbor, but park it in the middle of the peninsula. One thing catapults don't want to do is explore a narrow coastal strip when the sea is crawling with enemy warships. The other catapult heads towards the corner, and finds out quite quickly that you are on a fairly small island. Space will be at a premium. Fortunately, you won't need to build some of your low-tech buildings until much later, if at all. Produce a bunch of villagers and start sucking in wood. Pray that you don't get invaded, because you won't be able to deal with it.
There is a lion in the middle of your island. Don't let it hurt any of your units, but don't just kill it either. Render it down for its food, using at least three villagers, and do it right away, so that you don't have to worry about it any more.
Build a Granary and start sucking in food. It's not really safe to use your fishing boats much at this time. Work up to about 15 workers, including the fishing boats. If any of your units needs healing, get a priest.
Organize all 5 of your Triremes into one control group and clump them at the entrance to your cove. Set your diplomatic state to your enemies to neutral so that their trade boats do not suck your ships into their base. Build a Tower at the very tip of the peninsula. Pretty soon, naval attacks will start coming your way, but mostly it's only one ship at a time. Use standard naval tactics to blow the attackers away. Build a second Tower, and then a third one, on the peninsula protecting your harbor. Station two villagers there permanently to repair your ships and Towers. Get the missile weapon range upgrade at the Market so that your Towers are not outranged by the enemy warships. Research Coinage before pulling in a lot of gold. This is important here, because you may have an extremely limited gold supply. Coinage pays for itself whenever you have at least one untouched gold mine left, but is worth more the earlier you get it. Build two Towers on the far side of your island, in case someone decides to invade you from that side.
When you have a good supply of wood, get 5 more Triremes, and organize them into a second control group. This should put your population to 40. You won't need any more workers or ships. You will want a lot more priests, but don't produce any more units until you know what you can take over for free from your enemies.
With 5 Triremes and other assets to defend your harbor, you now have 5 Triremes ready to go exploring. Be very careful! Yellow has Laser Beam Towers (sic) that do 100 points of damage at a time, plus maybe 50% more for height bonus! Yes, they will hit a moving target! Yes, they can sink a Catapult Trireme with one shot! Fortunately, the reload time seems to be very long, and the range is only about 10, not the advertised 17, or perhaps the sight range of a Mirror Tower is less than the firing range. This means that if you are careful, your Trireme can escape after detecting a Mirror Tower. Also fortunately, not all the Yellow Towers are Mirror Towers. Once you have located a Mirror Tower, and it will always be the hard way, stay far away from it.
Brown has a small island base in the south corner of the map. There's not much there - an obsolete Tower, a couple of docks, and some Axemen. Yellow and Red appear to be trading there. Brown has no other ground units, and cannot produce any. There are a couple of small islands in the north corner of the map with some gold, food, and wood. The sea is crawling with fish. You could go fishing if you really wanted to.
The NE side of Syracuse appears to only have Ballista Towers. They only have a nominal range of 9 (Towers seem to have an actual range one greater than their nominal range), so a range-upgraded Catapult Trireme can take them out with impunity. There are also a bunch of other units there which must be dealt with as well. This appears to be the easiest place to attack Syracuse. Send a control group of Triremes and a couple of Catapult Triremes over there and start grinding away. The best way to deal with the many patrolling Elephant Archers is probably to shoot at one of the resting points in the patrol route with a Catapult Trireme using indirect fire. It will take a very long time, but eventually, all the Elephant Archers will be dead, and then you can close in and finish cleaning out the area.
While you're waiting for the indirect fire to have its effect, you might notice that Yellow and Red are collecting gold only very slowly. It looks like their only gold income is from trade. Send your unused Catapult Triremes to blow up the Brown docks. You are very unlikely to need that gold, and it might be useful to your enemies. In the meantime, the gold and stone mines on your island are probably exhausted by now. You should have a large stockpile of all resources. You may as well stop collecting wood, because you really don't want your woodcutters to open a hole in the forest wall on the NW side of your base. Yellow and Red appear to have gone completely static by now, and they don't even appear to be replacing their casualties.
By now you know that you can recruit elephants and heavy cavalry from your enemies, so you don't need to produce any yourself. You will need four more Legions, but you can recruit those as well. If those don't count towards the victory, then at the very end, you could simply disband as many units as you need to and produce your own Legions. Therefore the only units you would want to produce are priests, and maybe a couple of Ballistae, because they don't do splash damage. For the sake of argument, let's get a Scout as well, and a couple of chariots, because Yellow might have priests. With three production buildings available, you can get over the population limit by 2 in AoE by ordering a unit at each of them when you are one below the limit. This means that we can get 12 more units, even without disbanding existing units, and we certainly don't need 15 workers any more. Let's get 5 more priests, 3 Ballistae, one Scout, and 3 Chariots. But don't do this quite yet. Note that, when you are over the population limit, losing a critical unit can be very damaging, because in order to replace it, you have to throw a lot of other units away.
One way of wiping out the Mirror Towers that shouldn't be too expensive is to use range-upgraded Juggernaughts. Attack with a bunch of them. Whichever ones come under fire should have time to retreat, while the others stand and shoot. But as it turns out, you never need to do this.
After the NE side of Syracuse is clear of enemy units, go around the corner to the NW side. You need to kill all the archers and artillery, unless you want to convert them and let their ex-friends kill them. Most of these units are on the other side of a cliff, so you will not be able to extract them if you convert them. Push forward slowly along the NW side of Syracuse, eliminating all opposition. You should ignore melee units for now.
Load one Transport with 4 priests, and take along a villager for ship repairs. Take the second Transport along as well to extract the converted units. Send them over to the NW side of Syracuse and start collecting new units. Produce your required units first, of course. You will recruit mostly heavy cavalry at first, but as the terrain opens up near the west corner of the map, you should be able to get some elephants of various types. Remember, double-team each target with your priests, and go for groups of one or two units only. You will probably run out of space in your second Transport for all the converted units. You can unload them on the nearest island, but watch out for the wildlife!
Eventually, you will have cleared as much of the area around Syracuse as you reasonably can from the north, and you start to run into inland Towers. At this point, send your home defense Triremes to clear out the seas along the SW side of the map. Be careful, because there are plenty of Mirror Towers in this area. Send one of your Transports back home from the north, swinging wide of the Mirror Towers, of course, pick up the two remaining priests, a catapult and all the Ballistae, a villager, and maybe a few melee units, sail around the map edge, and unload to the SW of Syracuse. Push inland, killing or converting enemy units as you go, and blowing up Towers, link up with the northern force, and invest the land approach to Syracuse. Collect some of the plentiful resources over there if you are inclined that way, but you certainly won't need them.
Avoid scouting with your artillery, because if they find a Mirror Tower the hard way, it will be the last thing they do. Scout with heavy cavalry preferentially, because they can take one hit from a Mirror Tower and get away. They can get away from anything else that shoots at them, even if they can't see much more than what shoots at them. This is called reconnaissance by drawing fire. If you are really worried about finding two Mirror Towers at once, you can explore with elephants. Once you know that there are no Mirror Towers in an area, you can bring up your artillery.
There is very little opposition inside Syracuse proper. All the ranged units should have been killed already, and there aren't a lot of Towers, and none of the really nasty ones. Convert all the Legions you can, and everything else, of course.
There is very little chance of killing Archimedes by accident, unless you are doing something very wrong. He looks like a priest, but isn't. He can be converted.