If I have a Spright token on a battlefield and I play a Hidden Evelyn Entrancing as a reaction to the temporary trigger and charm a unit to this battlefield, does the Spright token die before the showdown happens?

Yes, the Sprite token is removed from the battlefield before the showdown occurs.

When the Temporary keyword triggers at the start of your Beginning Phase, it places a triggered ability on the chain. You may respond to this trigger by playing your hidden Evelynn, Entrancing as a [Reaction].

The sequence of events is as follows:

1. Beginning Phase: The Temporary keyword on the Sprite token triggers and is placed on the chain.
2. Reaction: You play your hidden Evelynn, Entrancing from the facedown zone as a [Reaction] to the Temporary trigger.
3. Resolution (LIFO):
   • Evelynn, Entrancing resolves, enters the battlefield, and its ability triggers.
   • You resolve Evelynn's ability, moving an enemy unit to her battlefield.
   • The Temporary trigger resolves, and the Sprite token is killed (sent to the trash).
4. Showdown: A showdown begins once the chain has resolved and units from different players are present at the battlefield.

Because the Sprite token is killed during the resolution of the Temporary trigger, it is no longer on the battlefield when the showdown begins.

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