I think some Egged services in Eilat run on Saturday as well. No.15Alef for instance between the hotels and to Taba at the border.

Then in the case of TA, there are the yellow Sherut taxis which ply specific routes within the city and only a bit more expensive than the buses. Better and much cheaper option IMO compared to the normal white taxis if you know where you want to go and don't mind being dropped off not at your doorstep.