There is no airport in Sparti. The closest airport to Sparti is the airport of Kalamata. This small airport works only in summer and receives charter flights from abroad. From Kalamata, you can go to Sparti by rental car in about 90 min.
Another convenient airport is the International Airport of Athens that receives flights from all Greek islands and from abroad. From Athens, you can go to Sparti by rental car (3 hours drive) or by bus from Kifissos Bus Station. From Athens, you can also organize your Athens airport transfer online.