To unlock Flying Mario, you must, sadly, have access to all the domes. Once you have done this, go to the Gate dome. To get there, go up, past the engine room, use the teleporter and you can go through the gate. You'll find yourself in the place where you caught the Luma rabbits at the beginning of the game. Go to the castle and speak with a red Luma. He'll tell you to collect 100 coins. Then you can use the flying star to do so, and collect the ones on the ground. Once you have done this, return to the observatory and look around on top of the bedroom, or near the terrace for a red star. Use it, and you can fly around for a while. (They respawn once you've used them).