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.

The shop also has a small, but well designed, climbing wall. The proprietor offers beginners courses which are generally held at Droushia/Ineia and Dhiarizos.
It's contact details are:

ONE STEP FURTHER LTD
18A Arsinoes Avenue
2006 Nicosia

Tel/Fax: (+357) 22 311012
Email: info@onestepfurther.com.cy

Map (opens in new window).

I'd recommend dropping them an email, or phoning, before visiting to make sure of their current opening times.