during beginning phase, before killing the sprite, can you react with a hidden card (which is sprite's call in this case) on the same battlefield the sprite dies?

Yes. The Temporary kill is a triggered ability that goes on the chain at the start of your Beginning Phase, which makes the state Closed — and a facedown card with Hidden gains Reaction while hidden (Rule 811.6), so you can play Sprite Call in response.

Because the chain resolves last-in-first-out, your Sprite Call resolves first and creates the new Sprite token, then the original Temporary trigger resolves and kills the old one. The new Sprite isn't checked for Temporary until your next Beginning Phase, so it stays.

One restriction: playing from hidden, the Sprite must enter at the same battlefield Sprite Call was hidden at — you can't redirect it elsewhere (Rule 811.1.d.3). That's fine here, since it's the same battlefield where the original sprite dies, and it keeps your presence there for the hold point.