cyprus airways fly from tel aviv to larnaca, 1 direct flight most days dep 07h00, returning from larnaca most days 22h30, flight time around 1 hr. picking random dates next month, return fare cyp148-00.
There are no flights between the Republic of Cyprus and Turkey. The only way you can travel by air from the island to Turkey is to go from Ercan in the illegally occupied north of the island, and you can only do that if you entered the island north of the "Green Line". As no country other than Turkey recognises the illegal regime in the north there are no flights to anywhere other than Turkey from Ercan, and no sea services to anywhere other than Turkey from the north either, so the only way into the north as a transit passenger is from Turkey - and you can only transit back to Turkey!. If you want to to go Israel-Cyprus-Turkey on a single trip you won't be able to do it. You could possibly do Turkey-Israel-Cyprus or Cyprus-Israel-Turkey.