One good possibility is O'Connell's Irish Pub at 3 Ben Shetach St which would be about 5-10 minutes from the hotels. It is a Kosher combination pub and sports bar which is very popular with the local crowd. It serves reasonably priced comfort food including great burgers.
The music ranges from "solid" at the beginning of the evening for the older set and changes by the hour as the crowd changes and it fills up
For another local posiibility you could walk to Agrippas St in the Machne Yehuda market (about 30 minutes) or take a taxi. There you can get the local speciality, "Mixed Jerusalem Grill" at any of the restaurants . Sima is the best known but it is the priciest and caters also to a lot of tourists. If you want local try any of the "hole in the wall" joints on the other side of the street which are really quite large inside and filled with locals