The nearest airport is Larnaca International, about 45 minutes west by road. Direct flights connect Larnaca to most major European cities. From the airport, you can take a taxi (expect to pay around 50-60 EUR) or catch a bus -- lines 711 and 711x run to Ayia Napa, and timetables are available at the airport arrivals hall.
Renting a car at the airport gives you the most flexibility for exploring the island. Prices are comparable to other Mediterranean destinations. Driving is on the left, as in the UK.