There is no airport in Gythio. The closest airport is located in Kalamata, the capital of Messenia. The airport of Kalamata is small and works only in summer. It receives flights from Thessaloniki, not Athens.
It also operates charter and low-cost flights from Europe. In fact, the popular low-cost air companies easy Jet and Ryan Air operate to Kalamata.
From Kalamata, you can get to Gythio with rental car or bus (via Sparti).