dont think it has a joe public airport but not sure – have only ever driven up :-\ has a landing strip but am sure this is only for air ambulances or private craft.
Inverness probably nearest but even then you would need to get train to closer – not an easy place to get to without a car but well worth it – have driven there to eat at The Three Chimneys – worth the drive – stopped over a night – been to skye a few times great for dogs, great for relaxing, walking etc but like I say public transport is only option if not a car.