Good thing there is internet nowadays and you can book everything online. In my line of work, I fly around a lot. It is clear that the company pays me everything, but I always book everything myself. From myself I can advise you the site where I book my transfers here https://gettransfer.com/en/directions/cyprus. You can choose any class of car and specify immediately where you need to meet you and where to take you. According to my experience I can say that a comfort class car is the most comfortable and inexpensive option.