I 1cc'd Mushihimesama Original this past weekend. You can do it too! Here's my advice for those fast tracking bullets and general tips:
I like to separate bullet patterns into two different types of situations:
Popcorn Enemies/weaker enemies w/fast tracking bullets:
1. Use large sweeping from left to right movements in a "smiley face/curve" motion to kill the popcorn enemies as soon as they appear on the screen with Auto Shot (C shot). They shoot out a lot of fast tracking bullets and your "smiley face" or even a figure eight movement will break up bullets and make it easier to dodge since the face tracking bullets are being pulled in wider angles. If there is a steady stream of fast bullets, tap dodge by gently nudging your stick/tapping keyboard key/pressing the directional pad in small movements. I like to tap in time with the music to keep rhythm. Also by sweeping and tapping you herd bullets. For instance I will move Reco in a curve to the left in a smooth motion, then quickly cut to the right within the space that curving to left had created.
Tough enemies which spit out slower, denser bullet patterns:
2. Kill the tough enemies with hard laser (A shot), but don't stay in the same place holding A shot. Tap dodge so you skew the bullets that are tracking you but at the same time, you'll be negotiating the predetermined bullet layer. Don't use the large sweeping movements (at first, until you are comfortable with the bullet pattern layer) within these tight patterns or else you'll hit one of the bullets. You'll notice that each tough enemy has a specific pattern to the patterns, make mental notes of how to move in response to each enemy's pattern and the sequence.
As you watch a bullet pattern, pay attention to the safe "rays." Hydeux makes a great point about not looking at the bullets themselves, but at the SPACES between the bullets. Check out the picture here:

I've drawn green lines to where you should imagine lines connecting the bullets. This helps you envision safe space and the rate you need to move Reco in between the green lines. It is much easier to envision patterns this way, instead of paying attention to each bullet.
More tips:
Stay at the bottom of the screen at first. Time your left and right movements by getting into the rhythm of where the enemies come out in waves. You should be thinking "okay, now I'm going here. Now here, now I sweep here, etc."
Watch your replays. Pay attention to where popcorn enemies come out and what side of the screen (kill those first). Pay attention to the large enemies' bullet patterns and their sequence. Practice a safe dodging scheme for each of these patterns.
Mushihimesama really grew on me and I hope this advice will help you get better at it too.
Cheers,
-the F-man