This stage is a royal pain, mostly from poor game design decisions.
Progress upward from the beginning of the stage, fighting your way through. It's not too hard at first. The problem is that there are invisible holes in the ground. These can be detected by sending a Bubble Lead out ahead of you. Or you can use Item 2 to get across. Better safe than sorry.
Fight your way through the horizontal corner and go down at the end. You'll encounter a series of lifts similar to Crash Man's stage, and they are much tougher this time. The Terrys are back, and there are spikes at the bottom that mean instant death. Use weapons to get through these screens, and be careful - you don't want to sleep on a bed of spikes.
Just when you thought the worst of it was over, after four increasingly-tough screens of the lifts, you'll encounter a floor filled with Sniper Joes - some of which are on mechas. Just fight your way through the only way you know how. And, of course, don't die. Make sure you have full Crash Bomb energy going into this boss battle (it's impossible to win without). If you don't, kill Sniper Joes to get some weapon energy.
At the end, you'll be in the Cannon Room. The cannons are the boss of this area. This is why the stage suffers from poor game design: You need Crash Bombs to destroy the walls guarding the cannons and the cannons. Nothing else works. And you don't have enough Bombs to do both. There are two solutions.
First, note that you don't need to destroy every wall, just the two that are completely sealing off a cannon. Use Item 1 or 3 to get around the others. Also, note that the Crash Bomb doesn't need to hit the target directly- it just needs to be close enough to catch it in the explosion. With that in mind, in a couple places it's possible to destroy two walls, or a wall and a cannon with a single Crash Bomb, placed between them. With strategic bomb placement, it's entirely possible to get through this in one try.
Here's the alternative method:
As a little tip, rapidly hit
when the cannons start blinking (this is the telltale sign that they will fire soon). Mega Man faintwarps when you go to the subscreen, so if you keep doing this as they fire, the shots will go right through you!