The shuttle from Fuerteventura Airport to Corralejo takes 40 minutes.
There is no direct bus service between the airport and Corralejo. However, you can take bus number 3 (stop on the 0 floor, runs every half an hour, price 1,24 EUR) from the airport to the capital Puerto del Rosario in 10 minutes, and there change to bus number 6 to Corralejo (also runs at half an hour intervals, ticket price 3,4 EUR).
Besides, near the airport you can take a cab (about 50 EUR), transfer (the price depends on the type of transportation) or rent a car for traveling around the island.
