I visited the Israel Museum, very good, unfortunately it closes at 2 pm (1400) hours on Friday due to the Sabbath. As a tourist this is very annoying and they are even stricter about Saturday opening hours in Jerusalem than Tel Aviv, nothing is open and no public transport either, some shops open at 9 pm (2100).

The City of David is an Archaelogical Park, basically ruins and some very long tunnels. Avoid the drainage tunnel, its about a kilometre long, uphill and slippery, if you are not claustrophobic at the beginning you will be by the time you get to the end. It is very narrow in parts, you have to turn sideways and low in parts where you have to bend to get through hahaha not for weight watchers.