Getting Door 4 consistently is the most important 'tech' in this level when you are starting out. Eventually you will get it everytime and focus on getting good bowser fights.
There are two common ways of doing the shot. The version presented here is the faster one, which is also harder to mess up once its learned. It's a little bit harder to learn though, since the other Version requires nothing but a single timed shot at a certain spot.
- Start the jump between the 2 red lines (visual cue in the background should be obvious)
- Do a FULL jump (Read as: You hold jump until you reach the apex of your jump) -
- NOTE: Do NOT start a flutter or it won't work (This is probably the only tricky part about this - getting a high jump without going into a flutter
- Right as you release jump you will start falling
- This is where you will start a jump
The green line roughly represents the path Yoshi will take.
The other strat is to run full speed right as soon as you enter the door and press A for a neutral egg at the right side of the 2nd window. You want to throw when Yoshi's body is underneath the line, not his head. The egg will miss door 4 if you throw too soon, and you'll mostly end up with door 3.
Swag shot: 
- Bowser takes 7 shots to be beaten.
- After the 3rd hit he will stop at how far away he is until you either hit him or he finishes his animation.
The delayed method is around 7-8 seconds faster than the consistent method and ~2 sec. faster than throwing ASAP.