Instructions
Go to the menu screen and set "Auto-Run" to "On." This will cause your characters to always run without you having to hold a button.
Enter the screen with the rat and follow him along the rafters to the left. Go down on the next rafter. Do not try to cut the corners when turning, but instead make a 90-degree turn. This way your character will not get stuck on the corner trying to get from one rafter to another.
Turn right on the next rafter and then go down to the second rafter you come to. Continue on this route going down, left, down and left. Exit the screen by going down on the next rafter.
Return to the previous screen and you should see the rat sitting on the rafter to your right. He is much closer than where he began.
Chase the rat in the reverse route going right, up, right and up. Turn left on the long rafter and repeatedly press the "A" button as you get closer to the rat. You should be able to catch the rat in the middle of the long rafter, and he will provide you with the computer code.