A good place to explore is the walk on the Old City Ramparts. You can enter the wall route at Jaffa Gate. The views are thrilling, but an entire circuit of the walls (about 4km/2 1/2 miles) is no longer permitted, because part of the route skirts the Temple Mount. Underneath the present Damascus Gate, the Roman-era gate, to the left and below, has been excavated. Within this classical, triple-arched gate (which may have been extant in Jesus’s time) there’s a small museum (closed at press time) displaying laser reconstructions of the original gate and worth a quick visit if it’s open.
Note: It’s not a great idea for anyone to walk alone on the ramparts at any time of day. The ramparts are at times patrolled by groups of unruly local kids and unsavory illegal “guides.” The circuit involves many ancient, irregular stone stairs, and although some have guardrails, this is not a place for young children or anyone not sure-footed.