You don't need to apply for the Schengen visa in Israel. You just need to make sure you can travel through Israel to get to and from the airport. Once you have your visa in place you will be fine. Be aware - you'll get grilled at airport security in Israel.
FYI - Ryanair and Wizzair have flights both from TLV and/or OVD (near Eilat, across the border from Aqaba) airports.