Any suggestions of what we can do from Friday night through Saturday?
What will be open?
On Saturday the Old City will be open as usual, except for sites in the Jewish Quarter. The Israel Museum, the Bible Lands Museum, the David's Citadel Museum and the Museum of Islamic Art will be open as well. Check websites for Saturday hours.
In addition non kosher restaurants will be open as well as restaurants in the Old City, East Jerusalem, Ein Karim etc.
Abraham Tours runs their transportation only Dead Sea/Masada/Ein Geidi tour on Saturday if you wanted to go down to those sites.
Lots of places are open, but jus people watching and a day of rest with time to catch up can be reviving.