The shared taxi service from Jerusalem to TLV airport ("Nesher sherut") runs 24/7 even on Shabbat, but you need to make a reservation before Shabbat. If you're staying at a hotel, have them do so. Yes, lots of flights leave late night.
There are a number of restaurants and bars open in West Jerusalem (the Jewish part, where most of the hotels are) even on Shabbat. You'll be able to have a nice dinner or even a drink on Friday night without trekking all the way to the old city. Don't worry; even though the vast majority are closed, Jerusalemites who don't keep Shabbat still manage to spend their Saturdays in bars and restaurants and cafes.