Can Fae Dragon's 'when you play me' trigger buff itself?

It can see and buff itself. 'When you play me' (and similarly 'when you play' a spell or Gear) is a trigger condition that means 'when you resolve' — it triggers only after the unit has fully resolved and is already in play, so by the time the trigger goes on the chain, Fae Dragon is a valid target for its own buff.