Use the ring on the wall to perform a wall run, you can do it from both directions, depends on which you like better (or which doesn't move the camera to a weird position.
What you need to do is get high as you can and when you are in the right position (when you reached the painted column) so a latteral jump towards the stone platform on the other side.
If you did it right, Lara should grab it.
Jump and grab the ledge on the wall, then jump towards the thin pole. You'll need to swing across quickly since it is a timed one.
Pick up the artifact #4 and the ammo.