There is no airport in Jerusalem. If you fly from Amman, you will end up at Tel Aviv (Ben-Gurion) airport, which is between Tel Aviv and Jerusalem. From there to Jerusalem you can take Nesher shared taxi (see post a few items down), NIS 50 (about $US 13) 24/7 to anywhere in Jerusalem.
There is some kind of public transport from Amman to Jerusalem - you may have to change somewhere. Other people know more about this than I do. No possibility of booking in advance. Just turn up.
If you rent a car in Jordan you cannot take it into Israel, which includes all of Jerusalem.