Getting from Ramallah to the Ben Gurion airport late Sat/early morning Sun?

The Nesher shuttle will pick you up from anywhere in Jerusalem. It needs to be booked the day before (book it on Friday morning for Saturday evening). The phone number is (02) 625-7227. Cost is around NIS 60 including luggage.

On Saturday evening, take Palestinian transport to the Damascus Gate, and hop over to some nearby landmark. I suggest the American Colony Hotel. Tell Nesher you want to be picked up there. (They would pick up from the Damascus Gate too, in all probability, but it's likely to be very crowded there and you might have difficulty finding each other with all the buses and taxis milling about as well.)

Tell Nesher the time you want to be at the airport and they will tell you what time to be outside on the street waiting for them. The company has been going for ever and is totally reliable. If it's a bit late it's because it's making the rounds picking up other passengers.

There is no direct bus from Jerusalem to the airport at any time. You need to take the bus to the closest junction and another bus from there, and in addition to that you would have to get across the city to the central bus station. With luggage. Why??!!

I completely second the opinion of Shuffaluff.
Nesher shuttle is reliable and will save you a lot of hassle.

I advise you to book a dorm in Jerusalem for the night. Take the 18 bus from Ramallah to Jerusalem in the evening, sleep there a few hours and take a Nesher sherut for the airport(needs to be booked in advance by phone).

The 18 bus does not run after 9 or 10pm so it's easier if you can sleep in Jerusalem.