The Old City is like Times Square in New York, there's nothing quite like it but if you spend too much time there you'll quickly tire of it. Remember the Old City is just a small part of Jerusalem and while it is without a doubt the main attraction the rest of the city isn't to be overlooked. I personally would stay in W Jerusalem. I like bars within close reach of where i'm staying so I can unwind at night just before I head off to bed and aside from a few places to get a drink around the Jaffa Gate area there isn't a lot to do aside from wondering aimlessly around a very quiet Old City. I think it would be better to make the short walk from wherever it is your staying in West Jerusalem to the Old City whenever you want to visit, you may appreciate it more.
If you do want to stay in the Old City though the action really isn't far away, just a short 15 minute walk. Whatever you decide to do enjoy, and don't forget to stop by Ramones just inside Damascus gate for a cold beer and to watch the most diverse flow of people you will ever see in your life go by, his hospitality is second to none.