Can I star Crossed a sprite token in the beginning phase by reacting to its own temporary trigger?

Yes, the timing works — but Star-Crossed needs two targets, so it often can't be cast at all in that spot.

Temporary is a triggered ability that goes on the chain at the start of its controller's Beginning Phase, which creates a Closed State, and Star-Crossed's [Reaction] lets you play it there (Rules 816, 813). Confirmed by FAQ #6571/#5523, which allow reacting to a Sprite's Temporary trigger. However, Star-Crossed requires both a friendly unit and an enemy unit as targets — if the Sprite is your only unit, or your opponent controls no units, you have no legal cast.

Note that bouncing your own Sprite doesn't save it: the token goes to hand and immediately ceases to exist (Rule 186.1), so it leaves play just the same. The difference is it isn't "Killed" — so any Kill/Deathknell-style triggers won't fire, and the Temporary trigger resolves afterward with nothing left to kill.