Welcome.

It may suck, but with Carthage you have to manage the best you can without archers. It makes their campaign a little more tricky, and means you need to rely more on infantry and cavalry, but that's no bad thing. If you take over Greece and the few Hellenised settlements in western Asia minor, you can then start hiring Cretan Archer mercs. They're very good, but obviously in short supply.

If you really want Carthage to have archers, you can download various mods. I have a free Carthy archer to download found by clicking on the silly blue banner in my signature, and I'm sure other people can point you in the direction of others.