The Jerusalem Hotel behind the East Jerusalem bus station to Ramallah, Qalandiya etc used to run an alternative tour to the wall, settlements etc. You could ask them about their drivers.
Alternatively, any of the taxi drivers lurking around Damascus Gate will take you anywhere you want. Probably cost though if you want to hire them by the day.
The wall is easily reachable from Jerusalem by bus or taxi and then you can just walk around it at will more or less.
Palestinian busses and Servis Taxis to the south and east(Bethlehem, Abu Dis, Beit Jala) run from the bus station on Suleman street opposite the walls of the Old City. While busses to the north (Qalandiya, Ar Ram, Shuafat etc) run from the Nablus Road bus station next to the Jerusalem Hotel.
Go to www.ochaopt.org to download the latest maps of the roads and wall.
The settlements around Jerusalem are also easily reachable by busses operating out of the Central Bus Station on Jaffa St.
www.egged.co.il/eng for timetables and routes.
There will be a bus taking in the northen ones, Psagot etc, and a few heading south to the Gush Etzion block, Har Homa, and Gilo.
Or you can hire a car yourself. Plenty of car hire companies on King David Street. I used to use a Palestinian one on the road to Ar Ram called Petra. Very cheap but I think they’ve closed down now.