There is no airport in Mystras. The closest airport to Mystras is the airport of Kalamata. This airport works only in summer and receives charter flights from abroad. From Kalamata, you can go to Mystras by rental car.
Another convenient airport would be the International Airport of Athens. From Athens, you can go to Mystras by rental car or by bus from Kifissos Bus Station (via Sparti). From Athens, you can also organize your Athens airport transfer online.