1. The Old City hostels — Inside the walls
Sleep inside the walls — the sights are a two-minute walk, not a bus ride.
A cluster of hostels operates inside the Old City itself, mostly in the Christian and Armenian Quarters near Jaffa Gate, in old stone buildings with vaulted rooms and, in several cases, roof terraces looking over the quarters.
The advantage is total: you are inside the walls, so the Western Wall, the Church of the Holy Sepulchre, the Dome of the Rock viewpoints and the souk are all a few minutes on foot, and you get the Old City early in the morning before the day visitors arrive.
The trade-off is that the quarters are quiet at night and you will be walking out for anything else. Rooms are old-building basic rather than modern.
Good to know: A roof terrace is the thing to ask about — it is the best feature these buildings have.