Jewel Man's stage is nightmarish for a number of reasons. Spikes and bottomless pits galore, a seemlingly-endless assortment of baddies, and a miniboss that will make you rip your hair out.
Early in the stage you'll encounter spinning ground-based green discs that descend from the ceiling, as well as mine cart-like foes. The discs are impossible to hit with the arm cannon, so you want to hit them before they land. The mine carts shoot small shots at you, and if you defeat them, they send out some "last" shots, so dodge those.
The spider enemies encountered throughout the level will drop from their threads if you stand too close to them, and they will then hop in an effort to flatten you. Best to destroy them before they can become too big a threat.
You'll also encounter platforms throughout the level. You need to shift your weight on these platforms by going from one end to the other, gaining momentum in order to make the jumps. Be especially careful in the jump to the miniboss' lair, too much momentum or too far a jump can give you a premature death on the spike wall above the miniboss door.
The boulder miniboss in this level is VERY frustrating. He shakes in the ceiling, causing boulders to rain down, which are very tough to dodge, and hurt severely. After a few fall, he himself will fall down. If you're on the ground when he hits (even if you're flashing with invulnerability from an earlier hit), you'll be immobilized. Counter this by ALWAYS being in the air when he lands. When he's fallen, rapid-fire him with the arm cannon (Plug Ball works especially well, but you probably don't have it yet.) He'll then roll back into the ceiling and repeat this pattern. Dodge the boulders to the best of your ability and don't get immobilized. Death in this fight puts you back at the beginning of the level.
The rest of the level gives you more of what you've already encountered, taken to the next level of challenge. At one point you must sway one of the platforms to jumo down the pit to the next area right beneath, dodging a whole wall and floor of spikes as you do so. Exercise extreme caution in this area, and don't jump unless you're sure you can make it.
Jewel Man himself is a pushover if you took my advice and defeated Galaxy Man first. He patrols the arena back and forth, shield of jewels surrounding him. He's big and tough to dodge with the shield on, and will jump the instant you hit the jump button on the controller. But Black Hole Bomb will due him in quickly. It will suck in all the jewels that shield him, and do severe damage. By exploiting his weakness and his jump pattern, he is extremely easy to kill.
Victory gets you the Jewel Satellite, a useful weapon that can both protect you and easily eradicate weaker enemies. Don't overlook its usefulness in situations other than boss battles, particularly in areas with a never-ending supply of weak enemies, as you can easily stock up on items with it.