Catapults are indeed the key to take cities when you do not have tech advance relative to the AIs. In my experience, it is very very important to adopt at least one of the +2 exp civics (Vassalage and Theocracy) at the war time in that 'Longbowmen era'. You will then get your new units with exp 6 or 8 from cities with barracks, which means your new units could have two promotions immediately.
Concerning the promotion of catapults, most of the cases they are used for city attack, so City Raider is the single most important promotion one should consider. I am very sure that City Raider is way better than Drill in a city attack, no matter who the defender is. Another handy promotion is Barrage, which will bring considerable collateral damage. With Barrage II, you make 50% more collateral damage on the suiside attack. Usually, I have a Barrage II catapult in my siege army. He will be the first martyr after the city defence having been worn to zero. Sometimes, not very often though, you will need your catapult to attack a good stack of emeny out of a city. In this situation, Drill is of course better than City Raider. However, I still prefer Combat to Drill. The last promotion needs some discussion is Accuracy, which add 10% bombard damage. I am not sure how to calculate this 10% extra damage, but from what I got in the game, a catapult with Accuracy makes about 2/3 more damage than the normal ones. I usually also have a such kind of bombard expert in my siege army as well. By the time you or the AI raising the war, most of the cities have a city defence more than 40%. The capital will usually be 60%. An bombard expert usually save you at least 1-2 turns to start the final attack. This at least 1-2 turns means less enemy enforcements, less unit supply cost, and less war weariness.
