As far as I'm aware, as long as you have a valid drivers licence from any country, you can hire a car in Cyprus.
You usually have to be at least 25 years old to hire many cars from rental companies in Cyprus, although some allow you to hire from the age of 21.
If in doubt, check with the hire car company before making your booking.
If you plan to explore the island, I would recommend you hire a 4x4 vehicle as, although the roads are in a far better condition than they were 10 years ago, you will still find some tarmac roads turn into dirt tracks in the mountain areas! A 4x4 is also essential if you drive on most "yellow" marked roads on the map, especially in the forest areas.
In winter, for example, these tracks turn to mud after heavy rain and are only passable in a 4x4.