When in Jerusalem, Mount of Olives and Mount Zion are practically unmissable parts of the itinerary, both for the historic and religious sites situated on them, and in case of Mount of Olives for the canonical wide perspectives over the entire Old City.
There are several interesting sights on the slopes of Mount of Olives, such as Dominus Flevit Church, Mary Magdalene Church, Church of All Nations and the Garden of Gethsemane, the Tomb of the Virgin. Mount Zion is headlined by the Tomb of the King David and the Abbey of the Dormition, as well as the Room of the Last Supper.
This tentative submission is meant as an extension to the already inscribed Jerusalem Old City, and I fear it will not get sufficient consideration with Israel having left UNESCO a few years ago. To me, these and other parts of the area surrounding the walls of the Old City are an indelible part of the historical and cultural complex.