There is no airport in Gythio. The nearest airport to Gythio is the airport of Kalamata that works only in summer and receives charter flights from abroad. Gythio is about 2 hours drive from Kalamata. You can go there by taxi or rental car.
Another convenient airport is the International Airport of Athens. From the airport of Athens, you go to Kifissos Bus Station and then take the bus to Gythio (via Sparta). The entire trip is about 4 hours. From Athens, you can also organize your Athens airport transfer online.