If an opponent plays Ocean Drake, can I Star-Crossed before it's resolution?

Yes — you can Star-Crossed Ocean Drake once its "when you play me" trigger is on the chain, and that's the only window you get.

A unit doesn't linger on the chain the way a spell does: during finalization Ocean Drake becomes a game object (on the board) and its "when you play me" trigger is then put on the chain, closing the state. That's when you can play Star-Crossed, since [Reaction] is exactly the permission needed in a Closed State. Star-Crossed resolves first, so Ocean Drake (as the enemy unit) plus one of your own units go back to hand.

The catch is that removing Ocean Drake doesn't stop its trigger — the ability is already a finalized chain item and resolves independently of its source. Since it's a "may," your opponent still gets to return a non-Dragon unit to its owner's hand (never Ocean Drake itself, it's a Dragon). Also remember Star-Crossed needs both a friendly and an enemy unit as targets; with no friendly unit on the board you can't cast it at all.