Getting an A rank is actually easier than it sounds.
The first thing you should do is collect 20 blue orbs as fast as possible and give it to the jellyfish. Now don't continue onto the next section yet, try to build up as many points with the time left on the clock.
The easiest way to get points is by getting as many links as possible without breaking the chain. Sometimes you'll come across a blue and yellow ring which sends you into a sort of a bonus time, when this happens quickly fly to the top of the stage and mash L1 and R1 as fast as you can and you'll build a ridiculous amount of points. That alone should be enough to get an A rank.
If you don't have an A rank already by the time you reach the boss it's not too late, simply beat the bosses as fast as possible to get the highest multiplier.
Here's a video that shows some good boss strategies. All credit for the video goes to the original creator.