A boat in a day from Fethiye is do-able, but will be expensive as the boat will need to be hired for 2 days (return empty)
Its about 280 Km by sea from Fethiye so sailing 15 - 20 hours (on a really good day)
there is no way to travel all that distance in one single day by boat...unless you can find a speed boat..and FF is right the cost would be very high.
If you are in Patara right now..why not take the bus to Cirali ..about 1 hour from Antalya and spend a night or two there to be closer to Antalya for your flight. Cirali is 90 minutes from the antalya airport.
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