Getting There

Use fast travel to get to the base camp Mongolian Passage. (On the map this is the second one from the lower left, south-west in the Siberian part.)

Solution

Head up the ramp on the left. Jump through the water curtain into the next passage. Step forward to reach the edge and jump to the next ledge. Pull up and walk along the narrow path. Lara will spot an ancient war galley stuck in the ice wall. Jump towards the climbable ice wall and scale it, using the axe. At the top head to the right, into the next part of the cavern. Scramble up the wall ahead and follow the path upwards to the left.

You will come past a spot where a narrow beam is sticking out of the walkway to the left. This is where you can reach the challenge tomb. Step onto it, to jump from here to the climbable ice wall. Climb up, using the axe. Run out onto the wooden planks and from here jump towards the mast. Grab the beam of the red flag and climb up, onto the mast. Balance across and at the end pull up. Now stand in front of the golden cog mechanism and use the axe to pry loose the hook holding them in place. A dangling brass kettle comes into focus on the right side. Scramble up the wooden board just behind the mechanism to reach the beam above it. Balance across it and out onto the mast. At the end jump towards the brass vessel. It will swing against the ice wall, breaking the ice that prevented Lara from climbing up the wall. Wait till you have safe ground below you, then let go of the vessel and run over to the newly exposed ledges on the right. Climb up using regular jumps and wall scrambles. At the top traverse to the left to pull up. Here you will find a crank. Turn it counter-clockwise as far, as you can, using the left analog stick. This will lower another brass vessel. You will now be faced with a simple time run. Let go of the crank and immediately run out onto the mast. Follow it to the right end and from here jump towards the kettle which will break the ice on this side. Again wait till it has reached its final position to drop down onto the ledge safely. Jump over to the next ledge, below the climbable ice wall and scale it, utilising the axe. When you reach the wooden beam, jump up to the upper beam. It will come loose, forcing Lara to secure her grip. Shimmy sideways to the right and jump over to the next wood beam. Traverse further to the right and pull up. Wall scramble up to reach the top. Then head up the stairs and open the secret codex. You will gain the Ancient Abilities ("Quickly shoot up to 2 successive arrows without the delay of redrawing from the quiver. Rapid fire arrows by quickly pressing and releasing RT after an arrow is first shot.") and the achievement Challenging, as well as 10000 credits. Now turn around and head towards the left edge of the chasm. Here you use the rope slide to leave the tomb.