This page explains how to find the Ree Dahee shrine and solve the "Timing Is Critical" puzzle.
How to Find the Shrine
From the top of Dueling Peaks Tower, look down at the base of the nearest Dueling Peaks mountain. You should see a ledge with a mushroom growing above it. Glide down to this ledge and follow the path as it goes between the mountains. Keep following this ledge until you find the Ree Dahee Shrine.
How to Solve the Shrine
First, step on the orange switch in the floor. The platform up ahead will tilt, making the ball roll down into the hole. A moving platform will come to where you are standing. Walk onto the moving platform.
In the next area, step on the orange switch and watch the ball roll, but walk off of the switch before the ball rolls off, and walk onto the tile to the left of the switch. You need to make the ball roll onto the lower platform. If you stand on the switch too long, the ball will fall down below (but a new ball will appear on the platform after a moment). When the ball is on the lower platform, it will roll into the hole. The moving platform to the left of the switch will activate, so stand on it to go to the next area.
Go down the ramp and step on the orange switch. This time, you have to walk off of the switch shortly before the ball rolls off. If you time it correctly, the ball will be launched upward and land on the platform on the right, where it will roll into the hole. Stand on the moving platform to go to the next area.
If you want to get the treasure chest, use Magnesis on one of the barrels down below (near the final hole that the ball rolled into) and put the barrel onto the switch where you were a moment ago. Then you can walk up the ramp to reach the treasure chest, which contains the Climber's Bandana. If you equip it, you will climb more quickly, which makes exploration easier.
From here, go down the ramp and go toward the area where you got the metal barrel. There is a ladder near the barrel that you can climb up to reach Ree Dahee. Check on the glowing box to get the Spirit Orb.