There is just one climbing shop in Cyprus, based in Nicosia (Lefkosia). The shop is well stocked, and you can buy everything you are likely to need, including shoes, ropes, harnesses, hardware, climbing clothing, rucksacks, etc.
18A Arsinoes Avenue
Tel/Fax: (+357) 22 311012
Google maps location
I'd recommend dropping them an email, or phoning, before visiting to make sure
of their current opening times.