The city of Sarlat does not have a dedicated airport. Depending on where you are flying in from, there are several viable flight options that will bring you close to this medieval jewel.
The largest cities in the vicinity are Bordeaux to the west and Toulouse to the south. Closer options consist of the Bergerac airport, which is a good choice if you are inbound from London or Brive-la-Gaillarde if you are flying in from Paris.
Both Bergerac and Brive are located an hour away from Sarlat by car or rail (TER Aquitaine).