There are lot ways to communicate in Cyprus.
You can find traditionally, telephone booths in the touristic areas with a prepaid card or some cents. For a SIM card, you can buy one from mini markets, most common are Cytamobile-Vodafone, MTN and Primetel. However, your SIM card will work here but with some extra charge.
Also, the most of public places like restaurants, cafes and hotels have their own Wi-Fi free network. If you don't have access to any free network, you can connect to wireless zone with a message from mobile phone. That will cost approximately 2€ per hour and are available from Cytanet and Primetel.