Instructions
Open the box beneath the drunken sailors. You may see the rat wandering around this location. Use the stick with the box, then use the string with the stick. You get the stick from the prison cell on Phatt Island and you get the string from a table near the elevator leading to the swamp where you got the "bucket o' mud."
Use the cheese squiggles with the string to bait the trap. These are the cheese squiggles you picked up from the bowl at the inn much earlier in the level. Wait for the rat to start eating the cheese squiggles, then pull the string.
Open the box and pick up the trapped and cowering rat.