If I stole a Sprite Fountain with Akshan, will it instantly play a Sprite token on my side?

No, stealing a Sprite Fountain with Akshan does not trigger it to create a new Sprite token.

Reasoning

1. "When you play me" is a specific trigger: The ability on Sprite Fountain is a "Play Effect." According to Rule 383.4.a, this triggers only when the card is initially played to the board from your hand or another zone (the process of playing the card described in Rule 350).
2. Stealing is not "playing": When Akshan’s ability resolves, he moves the enemy gear to your base and you gain control of it. Moving a card from one battlefield or base to another does not count as "playing" the card. Therefore, the "When you play this" condition is never met, and the triggered ability does not go on the chain.
3. No effect on existing tokens: Gaining control of Sprite Fountain means you now control that specific Gear, but you do not retroactively trigger its play effect. Any Sprite tokens already in play that were created by the original controller remain under their control, as the control-change effect of Akshan applies specifically to the Gear itself, not the tokens it previously created.

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