The nearest airport to the clinic and hospital is ERCAN AIRPORT in Nicosia. if you are traveling from somewhere other than Turkey, there is no direct flights to Ercan airport, all flights are directed through Turkey. If you wish for a direct flight, LARNACA AIRPORT will be a better choice for travel. Larnaca airport is about an hour away from Nicosia by car (depending on the traffic). In some cases the direct flight might even be cheaper.
After the surgery you will be staying for 1 night (or 2 nights depending on the surgery) at the hospital (this is included in the price the doctor has given you). But for the remainder of your stay in Cyprus you may wish to find an accommodation of your choice. There are many places either in North or South who offer great services. After the surgery you will be expected to come for a check up within 1 weeks time. So it might be a good idea not to travel too far away from Nicosia.
It is best to remain in the country for about 10 days. This gives you enough time for the consultation (45 minutes), getting the blood tests (results come out the same day) , having the surgery and coming for the check up appointment.