Ring Man's stage is tricky due to all the "appearing and disappearing" platforms. These come in two varieties - rainbow-colored ones and brownish-colored ones. When you step on one, you'll see how they operate. If you got the Rush Jet from Drill Man already, it can ease the platforming experience.
You'll encounter four mini-bosses in this level - two hippos and two ring robots. The hippos are suspended on platforms supported by a column, simply blow the column segments away with your regular weapon and unlead on the big guy when he's low enough. The column segments regenerate, so you're going to have to keep firing to continue delivering pain to the mini-boss.
The ring robot looks like a mass of rings suspended above some sort of electronic unit. When he shoots out the rings, his eyeballs beneath will be exposed. During this time, blast the eyeballs with everything you have. Using the Flash Stopper to freeze him in a vulnerable state is a good idea, leaving you open to give him an everlasting case of red eye.
Ring Man himself is challenging, but nothing you can't handle. Come into the fight with the Pharaoh Shot - and remember you can charge it up before the fight! Charge and fire shots, while dodging him and his ring shots, and you should be able to stick it out. Just stay on your toes, and use an Energy Tank if you need one and have one available to you.