Reimu always has a homing amulet shot type, not only does this make the stage segments a lot easier as well, it removes the stress of having to stay under the boss. Some of the later games have that shottype switch to a forward-needle attack when you focus (hold shift to slow down) though.
But yes, staying under the boss is an acquired skill. An important tip to all newbies is that you should learn to focus your eyesight not directly on your character, but about an inch or two up the screen from her, so you can see bullets and patterns coming at you before they, y'know, actually reach you, giving you far more time to react. Getting used to playing like that also makes it far easier to keep the boss in your peripheral vision. All of the earlier games (I can't remember when ZUN removed this feature, maybe in SA?) also have a red boss marker at the bottom of the screen, below the field of play. You can follow that without having to look farther up on the screen.
Also don't hug the very bottom of the screen, as you're just removing one of the four directions you can move to dodge bullets. Avoiding the bottom edge of the screen also helps keeping the boss in your peripheral.