The fastest way to switch from patient to hasty is by hitting the second help box in this level. It is slightly faster than switching on the world map. Do this by spitting an enemy into the help box. This is more relevant on the Japanese version on the game, however, as there is one less text scroll.
The hardest part of this level is the falling pollen section. There are many ways to navigate it safely, either by stopping and avoiding the falling pollen, spitting enemies at it, or using eggs. This video uses an egg route. In a real warpless single-segment run, you would want to avoid collecting any red coins in order to save time at the score tally screen (it is minor, though: it saves half a second to have zero coins).
1-3 is otherwise very straightforward. It is recommended to exit the level with at least two eggs that can be used for 1-4.
Star cloud collection idea:
The trickiest part of this level is probably the end of the cave, which is shown in the gif below. Make sure to hold down B for as long as possible before bouncing on the shyguy, or you will not bounce high enough.