There is a shortcut you can take and avoid getting the Uli Key. That way you will enter the puzzle area without it. Not though, that you may want to take it. If you do not mind a spoiler, later on, you will need to find and use a second Uli Key. If you still have the first one, you do not have to find the second. Then you can use the first one and avoid facing two nasty enemies that guard the second Key.

When the level starts, get out in the courtyard and climb up the ladder on the left hand wall and follow the passage that leads to the switch. You can throw the switch if you want to, but it is not necessary if you follow the shortcut (you may want to do it, though, because you may consider getting the Uli Key afterwards, as mentinoied above).

Now, as you stand with the switch on your right, go left and follow the passage that leads to a closed gate. Before you reach the closed gate, you come across two consecutive openings on the right hand wall. From the first one, you jump to the pillar ahead and then the ledge with the Rocket. The second opening leads nowhere, but it is actually the one that will help us with the shortcut.

Go carefully to the edge of this opening and look down and to the left. There is a snow-covered ledge there and this is where you need to jump (where the X shows), but the jump is a bit tricky.	Stand at the very edge of the opening and turn left to face the ledge. To manage positioning Lara at the correct spot, stand firstly at the middle of the ledge. There is a thin line separating the tiles on the floor, so you can easily mark the middle. Then take a walking step forwards. Make sure that Lara's right foot is exactly at the edge and turn slightly to the right, only that much that will permit her to jump outside without banging on the opening's wall.
Now, take a simple standing jump (do NOT press Action!) towards the ledge outside. As soon as you jump, press the left direction arrow to make Lara change her direction a little bit.	If all goes well, she will land safely at the edge of the ledge!
You can also take a running jump. Position Lara like in the screenshot and take a running jump towards the ledge.	Whilst in midair, press the left direction key, so as to land on the ledge and not onto the ground of the courtyard.

Proceed forwards, past the opening on the left and throw the switch (it is one of the two switches needed to extend the ladder in the button room). Then climb up the opening you bypassed before and you will find yourself at the other side of a closed gate. Throw the switch to open it if you want to, then go down the slide that is opposite the opening from where you just entered, and on the right of the switch.

You are now behind the big closed gates. Go to the other end to throw the switch that will extend the ladder in the button room and open the gates nearby. Get out in the yard. Now you can go and get the Uli Key if you want to, although you will not use it any more here. Proceed to the button puzzle room as described in the walkthrough.