It's not clear how this scenario simulates the Battle of Zama, except that maybe your enemy doesn't have much cavalry, but does have lots of elephants.
Let's do something totally different in this scenario. Let's build a Wonder! This is somewhat risky, because you have to commit a lot of resources that would otherwise be spent on the army. However, your initial situation puts an early Wonder somewhat in reach, and you already have a maximum army, although the force composition leaves something to be desired, and you don't have any static defenses. Still, defending your own position for a certain length of time should be easier than fighting your way through a fortified camp under severe time pressure, because your enemy WILL build a Wonder.
You start with only 4 villagers vs. 15 for the Carthaginians. If they started building a Wonder right away, you wouldn't have a chance, but they don't. The first thing you have to do is pump out the villagers. 12 more villagers cost 600 food, so you will need to bring in 100 more. You will also need 2 Houses. 100 food is not a lot, so put only 2 villagers on this task, and reassign them as soon as you have it. Display the game clock and your population data with F11; this is important here.
The next question is whether you should get Coinage before collecting gold. As a general rule, you should do this, but here we are concerned with minimizing time. If we only get Coinage later, then we will have to collect 500+100 gold without it, so we will lose 150 gold overall, which may or may not be significant. If we don't get Coinage, we will need to collect 500 gold to build the Wonder. If we get Coinage first, then we will need to bring in 600 gold, which means that we will need to mine 480, so we save the time to mine 20 gold. But you need to factor in the food cost of Coinage as well, and it is significantly faster to collect 20 gold than 200 food. Therefore we will not get Coinage until later.
The next question is whether you should build a Storage Pit right next to the stone deposit. It will cost 102 wood, which is approximately 9 wood collection trips. Right now, the travel distance to get wood is small, but the travel distance to get stone is not very small. You need to collect 500 stone, which is about 38 collection trips. Is reducing the travel time to almost zero faster overall than increasing the number of collection trips by 25%? It doesn't appear to be a huge swing either way, bearing in mind that we are only making an estimate here. However, having the extra depot has additional benefits besides the immediate ones, so build that Storage Pit.
The next question is whether you should upgrade your woodcutters. This will cost 240 food (24 collection cycles) and 200 wood (17 collection cycles), and improves your speed by 1/6 at best over 50 or so wood collection cycles, so it is clearly not worth it here.
Split your 16 workers fairly evenly over the three required resources. Don't focus on one and then the next in sequence, because then your workers lose time by getting in each other's way. Keep an eye on your stockpiles so that you hit the 1,000 mark for each one at about the same time. Don't overcollect, especially not gold.
As soon as you have enough to start a Wonder, run all your workers to the far edge of the map and get building. If you have done this phase of the game correctly, you should start your Wonder at about the 5.5 minute mark. At your current build rate, your Wonder should be completed at about the 11.5 minute mark.
Somewhere in the 6-8.5 minute range, Hannibal will start building a Wonder. Clearly, this makes a huge difference, but in any case, you have 16 workers on it, and he has at best 15. Therefore, as long as you start yours first, you will finish it first, and then you will win on time unless he wipes it out before then. Practically speaking, he would need to wipe you out in order to wipe out your Wonder. It's good to have the shoe on the other foot for a change!
Even if Hannibal starts his Wonder before you, you can still blow past him. You might need to disband some catapults and replace them with villagers. Of course, the later Hannibal starts his Wonder, the better for you. Let's assume that he starts his Wonder at 8.5 minutes. By then, you have a comfortable lead, and it only gets bigger. With 15 workers, his completion time will be T. Your Wonder should be about half done by then, so your completion time will be about T/2. You will need only 7.5 workers to match his completion time. If you use 10 workers for what's left, you will finish first by a considerable margin. There is no need to get your Wonder built as early as possible; it merely needs to be on the clock before his. Therefore, you now want to pull 6 workers away for other duties. Put two on food, and one each on gold, stone, and wood, or something like that, and one can build stonework. Don't collect more than 100 gold until you have Coinage. Check back in later, and keep pulling workers off Wonder building for other duties, as long as you get your Wonder built well before Hannibal. Remember, if you have 10 points left to build and one builder, and Hannibal has 100 points and 15 builders, you will lose the race.
After your initial workers have been allocated and new ones ordered, send your three Scouts exploring. Don't be fussy; just send them to the far corners of the map. You won't be able to micromanage the exploration much at this time, so stop any Scout that gets close to a red dot on the mini-map. Maneuver the Scouts when you have the opportunity; maneuvering your workers is far more important. Locate the two openings in the Carthaginian wall and camp one Scout at each one, while the third Scout explores the rest of the map. You can use your other units as well.
Organizing your troops gets low priority, but it needs to happen. Your catapults need to be shoved as far back in your base as possible, so that they don't accidentally blow up friendly units. All your other troops need to be used to defend the two walls that you are going to build, in particular the openings. Your cavalry might remain outside your walls to hunt down enemy siege weapons. You probably want to have some chariot archers to assassinate enemy priests, and more priests of your own to sweet-talk enemy elephants, and you probably should disband some of your catapults accordingly. In any case, you really don't want to have more than one or three catapults.
As workers become available, build static defenses. You probably want two openings in your wall. Each one should be two spaces wide. You make gates with good melee units with stand fast orders. You don't want wider openings for obvious reasons; narrower openings are too hard for your units to navigate.
The Carthaginians never really attack you, so you get an easy win.
Historical note: Scipio, called Africanus, defeated Hannibal Barca at the Battle of Zama in 202 BC. The two armies were fairly well matched, except that the Carthaginians had 80 war elephants, and the Roman cavalry was somewhat better than the Carthaginian cavalry. The Romans won for two main reasons. First, Scipio used a tactical procedure to neutralize the charging elephants. He opened gaps in his lines to let the elephants through, and then dealt with them in the rear of the battle. Second, the Roman cavalry returned to the main battle after routing the Carthaginian cavalry and hit the Carthaginian infantry in the rear and routed them too.