The first stage in Dr. Wily's castle really isn't terribly difficult, if you just take it at a leisurely pace. It might even give you false hope that the last leg of your journey will be a breeze.
From the start, head right over a series of ledges. Here, you'll find two of the top-spewing monsters from Top Man's stage, so climb the steps carefully and take them out as possible. From there, continue to the right to the ladder. Stand on the ladder and from there jump to the right to collect the energy tank before dropping into the pit.
You're now in an underground sewer of sorts. Progression to the right is hindered by shallow recessed lined by spikes. If this feels dangerous, don't worry; it's not. Just make leaps over the gaps, shooting the fish that will fly toward you from the right as you go. If you take a hit and fall on the spikes, you should be able to jump away before any permanent damage is done.
On the opposite side of the pool of water, you can leap out and onto dry land. From there, climb a ladder that leads up to the next screen. There, you'll find a 1-up icon sitting pretty behind some greenish walls. Switch to the hard knuckle to take out the walls, then drop into the opening to collect the icon. Now slide left and climb up that ladder, into the next screen.
Once more, you're at a screen with those lovely green walls, and you can again destroy them with hard knuckle shots. Here, you should see another energy tank resting on the high ledge above, so go ahead and grab it if you need it (though if you used the password provided earlier, you should already have a full reserve of them).
When you've collected any goodies that appeal to you, climb the ladder on the screen's left side into the next room. To the right, you'll see one of the projectile-slinging foes. Make sure you don't climb to the very top of the ladder. Instead, hang on it and use arm cannon shots to take out your foe. Now you can go ahead and crest the ladder, then from there hop up to the ledge to your upper right. Now slide under the low ledge there and press right so that you land on the lower ledge, then repeat the action to land where your opponent stood a moment before. Continue right and there will be two more of the fellows to take out.
The second of those projectile-slinging guys is standing on a high ledge you can't quite reach without the Rush Coil. Switch to the magnet missiles and fire as he's open, and two direct hits will finish him off. Then it's safe to use the Rush Coil to spring up to that ledge, and to climb the ladder into the next room.
Now you're in one of the most potentially dangerous segments of the level. There are disappearing/reappearing blocks here, over a pit of spikes. Not pretty. If you watch, you should get a feel for the first three, which lead conveniently up to solid ground where an energy capsule is resting. You want to hop up to the third block, then slide under the first barrier to where the energy waits. From there, watch the pattern again. You should get a feel for when a block appears directly under the ladder on the left side of the screen. When it's just about to appear, slide under the low wall that stands in your way, land on the block and quickly jump to the safety of the ladder.
Climb the ladder up to the next room, then walk to the right (you can spring up to collect the special weapon refills if you really want) and through the gate that leads to the area boss.
NEXT: Stage Two