I HAVE 3 NIGHTS AND 3 DAYS IN CYPRUS ON THE WAY TO EGYPT. WHAT WOULD BE A NICE PLACE TO STAY NOT FAR FROM THE AIRPORT WITH A QUAINT VILLAGE FEEL THAT IS CLOSE TO A NICE BEACH.
The answer depends partly on when you are arriving and also which airport!
If Larnaca airport, Protaras is a good option as close to Fig Tree bay.
If Paphos, try Pissouri which has a nice village atmosphere and good beach.
Protaras will be very busy in high summer as close to Ayia Napa (hedonistic lively resort and best to avoid in summer unless you want lots of nightlife). Pissouri will be quieter.
But to get a true village atmosphere and a taste of "old", "real" Cyprus you need to get into the hills - there are countless villages to choose from. But then you won't be near the beach.