The Sick Man is a Story NPC that the player can interact with in one of the huts on the Frozen Village.
The Sick man plays a role in the Shanks Puzzle, as when you receive the golden cup from Desert Island, collect water, and talk to him, he'll thank the player and say:
"Thank you so much! Please meet up with my son, I'm sure he will reward you."
Meeting up with the son of the Sick man, Rich man who is located at Pirates Island, will continue the Shanks Puzzle further.
You can collect the water In the secret cave, where the Ability Teacher is. One of the ice sickles in the cave drops water, Just go under it with the cup in your hand and wait, and it will eventually drop water into the cup.