Going off of Wallless Fury's solution. The course is timed but remember you have indicators that appear along the walls showing the countdown by position of the red square on it, so don't panic. (Some of you might now say, like me, ahhhh that's what those are for! ;p)
Recommendation then (though you would want to do otherwise) is to take it slow. The wall run jumps require precision and hurrying it would just confuse the controls. Press up and A to run up the wall, let go of the buttons, then press the direction you want to jump and press A again.
You may also get stuck on the levers since you pause on activation for the cutscene showing "puzzle" progress then need momentum to swing down again. Stay calm and hold right trigger, A, and up to start up again.
As Wallless said, the levers stay activated so even if the timer goes down to 0 you can go back, pull the lever again, and continue activation from where you left off (though you still have the rerun the obstacle course ;) ).
Take the time to look for appropriate ledges to progress on and you should do fine. (Last segment actually required a bit of backtracking, so don't be afraid to look behind you.)
You'll know you got the last lever when you've got to the ledge overlooking the mausoleum temple... thing. (And the count down to 0, if you stay there, won't trigger the reset animation.) Jump on that lever and a flight of stairs will appear, the door will open, and you can drop back down into the water to swim to your prize. Congratulations!