Orochium Shrine is located near the Snowfield Stable on the South Tabantha Snowfield.
Shrine Location
Orochium Shrine is located at -1636, 2641, 0239 near the Snowfield Stable on the Tabantha Tundra, northeast of Rito Village.
Courage to Fall
You can pull on the door to open it, but there are lasers shining in the path beyond the door. If a laser shines on you, the floor will open beneath you, and you will fall into the pit.
Instead, go around the side and defeat the Soldier Construct that is patrolling that area. Then go to the end and use Ascend on the column above you. Climb the ladder at the top, then crouch to go through the short tunnel. Then fall down below to the area where there is a Soldier Construct. Defeat the Construct, then go past the stairs and go around the corner to find a chest. Open it to get 5 arrows.
Then go up the stairs that you went past, and you can crouch to go safely past the horizontal laser, while avoiding walking into the vertical lasers. Defeat the two constructs in this area, then step into the lasers to fall down below. This time, you are dropped into a hallway instead of into a pit.
In this hallway, there are some lasers moving back and forth, as well as some that aren't moving. Don't let any of these lasers touch you, or you will be dropped into a pit. After you go past the lasers, use Ascend to go up to the ledge above, where there is a treasure chest. Open it to get a Small Key. Then use Ascend to go to the area above.
Go down the stairs and around to the other side of the room surrounded by bars. Check on the door to open it with the small key, and take the orb. Carry the orb to one of the moving platforms near the stairs that you just walked down, and stand on the moving platform while carrying the orb. At the top, put the orb into the floor switch. The bars will open. Take the Fanplane out of the room where the bars were, then attach the orb to the top of the Fanplane. Then place the Fanplane in the rail in the floor and stand on the plane. Hit the plane to start the fans. When it reaches the wall, you can hit the plane again to stop the fans. Detach the orb from the plane and place it in the floor switch in this area. The bars will open. Go through and check on the dragon ring to get a Light of Blessing and complete the shrine.