There are two airlines that fly from Eilat to TLV and SDV: Arkia and Israir. If you are connecting to a flight out of the country you should fly to TLV. Sde Dov airport is at the north end of Tel-Aviv and it will take you around an hour to get to TLV from there, either by taxi or a taxi to the University train station and a train to the airport.

Either way you should arrive at TLV three hours prior to your departure.