There is an entrance exam for these schools, think it was in late Spring time. Best to give them a ring direct, though they will be on holiday now. Not sure about their admission policy, is it The Grammar School / Grammar Junior school or The English School? I am a teacher of 14 years and have lived an worked in Nicosia for 2 of those, it's just a good idea to go and have a look for yourself and see which ones suit the types of kids you have.
How are the job plans going?
Good luck.