The sack of Carthage! (didn't quite go to plan!)

[athanaric]

The key to winning siege battles like this one is

a) bring enough missiles, preferably javelins (since arrows are only moderately effective vs. armoured troops) and

b) patience.

Try to surround the whole town square with missile or precursor units with fire at will turned off (so as not to provoke the enemy to charge from the town square). If you've completed the maneuvre, you just need to micro manage the missiles to ensure they always hit the enemy in the back.
The top priorty missile target should be Phalangites. Spartan BGs can be cut down with AP melee weapons, if need be. Seeing as native Roman untis rarely have AP melee weapons (and you need those pila to impale the Elite Pikemen), try to hire Numidian archers, Germanic clubmen, Illyrian Levies or some Iberian troops to attack the Spartans (or Sacred Band, should they be present) from behind (while pinning them with spearmen or Roman infantry). Alternatively you can just use the expensive but awesome Extraordinarii.

How come Xantippos gets Spartan bodyguards?

He's a scripted Spartan who helps the Carthaginians against Rome.