How to Reach the Shrine
You will need a lot of stamina to reach the shrine. Cook some stamina-increasing foods, or just pray at a Goddess Statue to use your Spirit Orbs to increase your Stamina Wheel.
Then go to Akkala Ancient Tech Lab (or if you can't warp there, go to Katosa Aug Shrine and go northeast to reach Akkala Ancient Tech Lab).
Go into the lab and go up the stairs, then find the path leading to the roof of the building. Climb the tower that sticks up near the small end of the telescope. From there, look northeast to see the labyrinth. Float there with the Paraglider, using stamina-raising foods if needed. This is Lomei Labyrinth Island.
There is a Guardian Stalker in the first area of the labyrinth. It can't go into the narrow parts of the labyrinth, so if you don't want to fight it, you can run into the narrow passageways.
If you want to see the map of the labyrinth with the hidden passageways marked, click the buttons below.
About the Shrine
The treasure chest in the shrine contains a Barbarian Helm.
After the Shrine
When you exit the shrine after you go in and talk to Tu Ka'loh to get the Spirit Orb, there will be a hole in front of the shrine with an updraft coming up. Fall down that hole and you will reach an area with a bunch of Decayed Guardians and some Guardian Stalkers. You can use the Camera Rune to see which Guardians are dead and which will attack. There is a chest in the very center of the room that contains a Diamond Circlet, and if you purchased the Breath of the Wild Expansion Pass, there is a chest against the south wall of the room that contains the Travel Medallion.