After entering the shrine head for the large door and pry it open.
Go to the crank on the right side of the room and use it to open the hatch at the middle of the floor.
Now make your way to the left side and use the crank there to pull the cage up. Then head back to the right and use the crank again to close the hatch.
Go towards the door that you entered through and fire a rope arrow at the cage, pull it with the ascender and it should destroy the rocks on the other side.
Now go back to the crank on the left side and lower the cage again, but since you closed the hatch it won't go any deeper. Use the ramp behind the crank and climb up to the higher platform.
Pull the cage from the middle of the room and jump on top of it as soon as it gets close to you. Make sure to jump a little on the left or right so you're not in front of the chain or it will block you later.
As it swings towards the opposite end jump towards the wall and press space again to grab the ledge at the top. Drop down on the other side and you'll be back in Himiko's tomb.
Turn left and head inside the door that's open. The tunnel is pretty long but keep going until you reach the campsite. Upgrade all you can upgrade and take all the skills you need. If you still don't have it, please, please, please, get the penetrating arrows.
After upgrading all your stuff open up the door by prying it. Then head out into the snow.