Is there a bus service from Eilat to Jerusalem on Saturdays?

Yes, There Is

Yes, there are 2 buses on a Saturday night via Egged bus line #444: One bus leaves Eilat at 16:30 p.m. and a second one departs at 19:30 p.m.

It is a 5-hour trip; the earlier bus is due to arrive at Jerusalem's Central Bus Station at 21.30 p.m.; the later bus is due to pull in at 00.30 a.m.