Do you still have to pay the power cost when revealing a hidden card?

No, you do not have to pay any costs when revealing and playing a card from its Hidden state.

According to the RiftJudge FAQ (ID 4052 and 5952), when you play a card from Hidden, you pay nothing. The cost to "Hide" the card is 1 Power (by recycling a rune), but once it is in the facedown zone, playing it later as a reaction is free.

As stated in the FAQ:

• To hide a card: Pay 1 Power (by recycling a rune).
• To play from hidden: Pay nothing (the card is played for free).

Additionally, Rule 421.3 confirms that cards played from the facedown zone have their permissions defined by the effect that put them there, and the Hidden keyword specifically allows them to be played for free on subsequent turns.