There is a flying club at Akrotiri (LRCA) but I'm afraid I don't have any contact details. However, the controller at Air Ops (00375 527 6664) should be able to give you a contact name and number. If he gives you an extension as opposed to a full telephone number, substitute 6664 with the given extension. [00375 is Cyprus, 5 is Limassol area and 27+Ext is the direct dial no for Akrotiri]. Where are you going to stay when you're there, and have you been there before?
Have a great time, and not too much of the dreaded Red Infuriator!