I am a former road safety instructor, a professional driver and a member of the institute for advanced motorist, i have driven in Cyprus several times with over 30 yrs behind the wheel, i have on several occasions only just avoided local drivers [most don't drive they aim]
from protaras you can get a bus to Agia Nappa or up to Paralimni where you can then change for Lanaca bay or Lefkosia, there are also many trips available and local taxes are reasonably priced,,
if you do hire a car watch out for speed cameras and don't drink, only one pint is enough to put the average man over the limit {its one of the lowest in Europe}
enjoy your holiday
wizard