Jump across the water until you reach the shell enemy further ahead. Hitting it again will kill but you can just leave it there if you want. Hit the second shell enemy on the higher ground and then jump into the boat. You'll drift on open water for a bit and then a pirate ship will come into view. The open ports along the side of the ship will drop cannonballs when you pass under them.
One way to do this part is to stand on either end of the boat and walk when you pass under the porthole. The other way to do it is to just stand there and then move when you see a cannonball. Either way is fine. There will be a series of platforms leading to the top and a chunk of ground on the right but that doesn't go anywhere. Carefully make your way to the top and walk left.
There's a simple way to finish this stage, although you'll have to forget about the special item. Stand on the edge of platform where the mast is, jump and throw a star to the left. Jump onto the star when it comes back and then jump off onto the platform on the far right. Walk one screen to the right to meet the boss. Simple right? If you don't want to do that, head down the tube ahead.
Once you're inside the ship, you'll see several Ohagi enemies with cannonballs on their heads. Stand on the edge beside the tube and hit the Ohagi on the right with a star. Move onto the steps and hit the two Ohagi enemies walking around here. Head to the left and stand on the right side of the platform with the Ohagi on it. Now, toss a star at the other Ohagi on the barrels, then hit the Ohagi on the platform.
Interestingly, one Ohagi is completely still in this area and can be played with the second controller, although this serves no practical purpose. Either way, enter the tube to go topside again. Further to the left, a curious looking green enemy will come running at you. Fire three stars at it and jump over it as necessary to defeat it. You can also walk under it if you want. Keep going left and push the large cannon off the side of the ship onto the small platform below.
Stand on the edge of the large cannon and wait for it to fire, at which point walk left a little so that you end up on the cannonball. I suggest that you save the game here if you're using an emulator, since this part is tough to do. Once you make it to the far left, enter the tube and grab the special item. Walk further and you'll see an odd bird named Tori-san (トリさん?). Wait for Tori-san to start flying and then jump onto it to get taken back to the ship.
Walk to the right and hit the two Ohagi enemies here as well as the flying Ohagi enemy... or just avoid it and continue on. A few more Ohagi enemies will pop up and another bird will appear on the mast on the far right. Jump onto Tori-san, grab the Orange Medicine and fall off the side. The boss is just ahead so walk right to meet him.
BOSS: Kaizoku (カイゾク?)
Here's a fairly tough boss. This pirate has a sword that blocks your stars from the front so you have to jump around and toss them at him to score a hit. It will back up and rush toward you depending on which direction you're moving. As you continue to hit the boss, it will start to alternate between jumping around and rushing toward you swinging its sword. It takes six hits to finish it off. If the player gets to this spot earlier than usual, they will find this boss sleeping and an Ohagi acting as an alarm clock - pushing Kaizoku off the edge while it's drowsy will prevent the fight, although it is very hard (but not impossible) to reach the boss in this state if the player goes out of the way to obtain this level's magic item.