In the Old City there are numerous museums. A must-see is the Kosel tunnel tour. I believe there is also a tour of the southern side of the area of the kosel, it’s to the right side of Dung Gate. In the Jewish Quarter, there are houses that were found from the times of the Second Bais Hamikdosh and they have found mikvaos and beautiful mosaics, etc. There is also a museum where they have samples of the keilim that were used in the Bais Hamikdosh.
In Jerusalem proper, there is the Israel Museum and the Dead Sea Scrolls. To visit kevarim, go to Har HaMenuchos. There is a chelkas harabonim there with many gedolim buried there.
Also don’t forget that everything in Israel is close. There are buses and tours going out from Yerushalayim to all parts of Israel, including Tzefas, Amukah, Tiveriah, Haifa, etc. If you are here in June/July, there may be an erev Rosh Chodesh that you will be here and there are tours to the kevarim of tzaddikim in the north that day. Look at signs at bus stops, particularly if you are in the frum neighborhoods. Signs will be up the week before. Reserve a spot if you’re interested.
Enjoy your visit!