Famagusta is in Northern Cyprus, accessible from the north (Ercan Airport near Nicosia, with connections from Istanbul and Ankara) or overland from the south via the crossing points that link the Republic of Cyprus to the north. Crossing between north and south is now generally straightforward — you can cross on foot or by car at several checkpoints, including the Ledra Street crossing in Nicosia and the Metehan/Agios Dometios crossing.
From Ercan Airport, Famagusta is about 80km east, roughly an hour by road. Taxis and minibuses operate the route. From Nicosia (Lefkoşa in the north), there are regular dolmuş (shared minibuses) to Famagusta throughout the day.
The Republic of Cyprus's airports at Larnaca and Paphos are an option if you are entering from the south and crossing north — Larnaca is about 50km from Famagusta. Be aware of the political sensitivities: hiring a car in the Republic of Cyprus and driving it north (or vice versa) requires specific insurance arrangements.