Can't help you with Eilat, but in Jerusalem or other towns, check out the "street food". You'll find many little open shops where you can buy falafel and other local specialties.
I practically lived off falafel for 2 weeks when I was there. They'll hand you a pita with 3 or 4 fresh-fried falafel balls in there, then there will be an area where they have 10-20 different things you can stuff in there; hummas, sauces, marinated veggies.
For breakfast, your hotels will probably have a type of "buffet", and as long as you like fish for breakfast, those were always worth it (and came with the price of the hotel).
Ask locals for suggestions, and "go native" as much as possible.