The Egged bus company has buses from Ben Gurion Airport to Jerusalem as early as 6:20 am. There are also shared taxis, called "sherut" that are very affordable and run between the airport and Jerusalem at all hours of the day. They leave from the taxi stands at arrivals at the airport.
There is no regular bus service from Jerusalem and Amman. There are private bus companies that have service from Nazareth to Amman.
Have a great trip.