If I have a hidden Zhonya's with a Sprite unit then it dies when it's my beginning phase. Can I react to Sprite unit's temporary and play Zhonya to base?

The Short Answer

Yes. You can reveal a hidden Zhonya's Hourglass to save a Sprite with the Temporary keyword during your Beginning Phase.

Detailed Reasoning

1. Status of Hidden Cards: A Zhonya's Hourglass functions as a mandatory replacement effect. To utilize this effect, the card must be revealed and in play.
2. Timing of Temporary: The Temporary keyword triggers at the start of your Beginning Phase. As a triggered ability, it is placed on the chain, which opens a window of priority for players to act.
3. Priority Window: Because the Temporary effect uses the chain, you receive a window of priority before the effect resolves. During this window, you may reveal your hidden Zhonya's Hourglass. Once revealed, the replacement effect is active and will intercede to save the Sprite when the Temporary effect resolves.
4. The "Death" Event: As noted in RiftJudge FAQ #9663, Zhonya's replacement effect functions to replace the destruction event. By revealing the card before the Temporary effect resolves, you ensure the replacement occurs before the death is finalized.

Summary of Mechanics

• If Zhonya's is revealed during the priority window: Because it is a mandatory replacement effect, it will intercede before the Sprite dies, saving the Sprite while consuming the Zhonya's (per FAQ #10136).
• If the effect resolves without intervention: If you pass priority and the Temporary effect resolves without the Zhonya's Hourglass being revealed, the Sprite is destroyed and placed in the trash, at which point it is too late to save it.

Note: Be careful, as losing your only unit on that battlefield (the Sprite) at the start of your turn may cause you to lose control of that battlefield during the subsequent cleanup step, which would result in your hidden Zhonya's Hourglass being discarded (FAQ #7086).

Note: I could not fully verify this ruling against the retrieved rules.