I played Unchecked Power without a showdown, opponent had a Zhonyas Hourglass hidden at 1 battlefield, is he still allowed to save that one unit there?

Yes, your opponent is allowed to save that unit.

According to the FAQ regarding Unchecked Power and Zhonya's Hourglass:

• Unchecked Power deals 12 damage to all units at all battlefields simultaneously.
• Because the damage is simultaneous, all units that would be killed by this damage are considered to be dying at the same time.
• Zhonya's Hourglass is a replacement effect that replaces the event of a unit dying. Since the units are dying simultaneously, the controller of the Zhonya's Hourglass chooses which unit's death event to replace.
• Hidden cards can be flipped at any time you can play a reaction. When you play a spell like Unchecked Power, your opponent has an opportunity to react to it before it resolves. They can flip their hidden Zhonya's Hourglass during this reaction window.

Sequence:

1. You play Unchecked Power.
2. Your opponent has an opportunity to react (this is when they flip their hidden Zhonya's Hourglass).
3. Unchecked Power resolves, dealing 12 damage to all units.
4. The units that take lethal damage would die during cleanup.
5. Because Zhonya's Hourglass is a replacement effect, it replaces the death of the chosen unit, saving it.