To obtain the Princess Zelda Kidnapped side adventure in The Legend of Zelda: Tears of the Kingdom, speak with Penn on the roof of the Dueling Peaks Stable during the Potential Princess Sightings side adventure. To reach him, use Ascend in the stable!
For this quest, make your way to the location shown in the map below.
Once there, you’ll find Zelda in a cage.
Use Ultrahand to lift the cage releasing Zelda. Once you do, you’ll have to fight 3 Yiga Footsoldiers. Take them out to complete the quest!