I was watching a playthrough, and I saw this funky movement. He gets up the ledge without jumping/climbing.

This movement might apply elsewhere. It only appears to save less than a second, but if it can be used in multiple places in the run... every second counts.

Stupid share, but sharing anyway :)

Good eye! These are usually called pop-ups, and they do in fact save time in certain instances like you said. They're usually found at corners or edges of ledges, with your example being the corner where the two walls meet. While optimal in theory, often times it's quite precise to actually land on the floor above after getting the pop up, so these can be hard to save time with consistently in an RTA setting. However, you also have the option of jumpslashing after the pop up, which is easier and still saves some time over climbing, just a bit less. You'll see pop up + jumpslash very commonly done in the Dampe race with sword as an example.