Use the Rotten Fish to help the man under the bridge to capture the Kappa. If you try to go down the bridge, you will find a man on a boat, he like to capture the kappa but to capture this you will need a Rotten Fish. But helping the man will not give you anything in return. But why expect rewards for helping others?
How to Get the Rotten Fish
Go across the red bridge, and enter the next room. Walk all the way left until you spot a rotten green fish sitting on a large staircase. Looks tasty!