I think Gileum is right. Staying close to the left side of the screen moving up a little when the slow blue center shot comes. Move back and forth when the rainbow coloured lasers appear but stay in the middle. And retract to the left most side of the screen when the little bombs appear.
BUT ... this is easy to find out on the flash screen with an indistructable mouse pointer over the image. It is way WAY harder in a real game when every little move of your ships means »Game Over«. I also like the music (it's take from a real game, isn't it?) and the use of sound effects.
This would certainly be an awesome boss in an actual Gradius game. Certainly a difficult one, but I could still see ways to defeat it. Heck, I think one good strategy would be to stay close to the nosecone most of the time and keep your option units in front of it.
Stay in the middle, farthest back on the screen. Move up slightly when the central short blue laser comes your way. Unless something comes down the center that forces you to move more than a few inches, this boxx would be cake, no matter how flashy and laserific it is.