The first screen is mostly straightforward platforming with one small trick at the end. Instead of using the boulder at the end of the room to get to the door, it's faster to take a shell with you, spit it out in the hallway leading up the door, and bounce off of that and up to the door. The optimal strategy for this involves spitting the shell upward instead of forward, as is done in the video.
In the second screen, it is slightly faster to skip the cloud and clear out obstacles using eggs but it is much safer to take the cloud, as getting hit even once probably means losing time compared to taking the cloud.
Getting the key can be done by either pushing the crate, or hitting the red switch and ground pounding the crate where it spawns. It seems easier to hit the red switch than to use the crate pushing strat shown in the video below, and seems equally fast.
During the train section, push left/right during any of the first few forks in the track to turn the train around. This changes the spawn of some of the enemies along the track, letting you take the quickest route through the area.
In the final area, beware the red coin shyguys despawning if you move them offscreen (such as if you get hit).