Jerusalem is no longer divided (it was divided between Jordan and Israel between 1948 and 1967). Israel controls all of it, and there are no check points or border control anywhere in the city.
There are security checkpoints at the entrance to the Temple Mount, but they're there to keep the Israelis (and weapons) out, not the foreigners (as opposed to what you might have heard, even though Israelis control all of the city, it is the Israelis, Jews in particular, who are banned from the Temple Mount). Most tourist can get in without a problem. Dress accordingly. On Friday's, religious holidays or when there're some occasional security tensions and alerts, some restrictions may be in place (usually for the Israeli/Palestinian Muslims coming to pray, but may also affect tourists).
The yellow capped building is not the Al Aqsa mosque, its the Dome of the Rock and it's not a mosque at all. Al Aqsa mosque is the gray and unattractive building in the corner of the Temple Mount compound, right on top of the Western Wall.