Old Damascus is walkable if you do not try to see everything. We usually start near the Umayyad Mosque, then drift into Al-Hamidiyah and the lanes around Straight Street.
At the mosque
The mosque visit has a dress code. Shoulders and knees covered, and women will need a head covering inside. We say this before you go in so nobody is surprised at the door.
Al-Hamidiyah and the lanes
Al-Hamidiyah is a covered market. It is loud, it is useful for a coffee or a small purchase, and it is easy to get turned around. Stay with the group unless you have agreed a meeting point.
Leave time to sit. The courtyards and side streets are where the day actually lands, not on a checklist of names.