In a single-segment run, entering this level with at least one egg is important to save time hitting the balloon carrying the bouncy arrow faster. Most of the rest of the level is straightforward, with small optimizations here and there like jumping as soon as possible while riding the slow yellow lift.
Faster arrow show at the start: 
Near the end of the level, an extended flutter can be used to skip the beanstalk, saving a few seconds. There are many ways to get the enemy to land on the mushroom platform but an upspit as shown is likely the simplest:
Acquiring as many eggs as possible now helps later in 1-8.