In Pallidus' 120 sign run when he is finishing the seventh floor of the tower, he does a trick where you get shot up fast enough to trigger the sign without waiting for the block to travel all the way up. I accidentally replicated this once simply by walking forward, but have failed in trying to replicate it several times since. Is there a certain way to do it?

He does this around 4 minutes and 50 seconds.

It's unknown. You basically have to be lucky with your landing location and where the cube is when you land. Then you trigger it by either walking forward, walking forward and jumping, walking into the corner, or walking into the corner and jumping, but the setup is unknown and purely luck-based right now.