There is plenty to see in Jerusalem on a Saturday so it is not a factor in planning your itinerary. The Muslim and Christian quarters of the Old City are open and busy as usual. The Tower of David Museum near the Jaffa Gate is open. In the New City you can visit the Israel Museum (you can spend half a day there), the Bible Lands Museum and the Museum of Islamic Art. Check websites for hours.

On a weekday you can go on the Western Wall Tunnel Tour and the City of David tour - but note that these tours (highly recommended) must be booked in advance.