You'll start on a bunch of clouds moving slowly to the left. Jump from cloud to cloud until you reach the airship. A Cannon will appear and start shooting cannonballs. Kill it and the next one on the lower platform ahead. A Grim Reaper will appear above. Switch to your spread fireballs, jump and hit it. Avoid its swoop attack and continue hitting it until it dies. Now, kill the Cannon on the ledge ahead and then continue onto the clouds again. You'll come to the another airship soon enough. This time a Cannon will be waiting at the edge. Pummel it from the clouds and then jump onto the airship. There will be a Heart ahead, followed by another Grim Reaper and a Cannon. Kill the Grim Reaper first, grab the Heart, and then take care of the Cannon on the lower platform.
One final Cannon will be walking around on the higher platform just ahead and then more clouds. Continue cloud jumping until you reach another airship. Jump onto this one and kill the Red Robot running at you, or just jump over it if it gets too close. Following the Robot is a Knight. You can kill it or just jump over it. Further ahead, there will be a hole. Don't jump into this one, jump into the next one to find a Heart Container. Use the Bat Transformation to get out of this hole and into the first one you found. Continue ahead and kill the Blue Robots running around if you can. If you don't a whole bunch more will appear. There will be another hole ahead. Fall into it and then continue killing the Blue Robots to the right until you find the elevator.
Use it to take you back on top of the airship. There will be another Red Robot waiting for you right beside the elevator so kill it quickly. Use the Bat Transformation to get onto the wall ahead and then kill the Red Robot running at you, followed by another Knight. Jump into the hole and then DUCK. I'm not kidding. Soon after you do, a huge energy beam will come from the left and obliterate the Blue Robots roaming around. After it disappears, jump out and continue to the left. Jump into the hole at the far left when it's clear and kill the Blue Robot at the bottom. Grab the Heart at the right and fall down again and then continue to the left to find the boss.
Boss: Giant Robot
This boss is simple as long as you have enough health. Use your Upside Down special to stick to the ceiling. Fire at the Robot's head as much as possible and just take the Robot's hits. After while its arm and then the rest of its body will explode and disappear.