Here's the quickest way out.
After the first room (with the note about pipe usage), go down (and left) to the next pipe. You should be in a room with a big pool that has a pipe underwater. Go the the pipe on upper right. Go in and you should be in a room with ghosts, a trampoline, and a ! Switch. Hit the switch to drain the water. Go back one room and the pool is drained. Go in the pipe that was underwater. This room has a Save Point. Go in the next pipe and you'll be at the end of the maze, where Belome awaits.
"No way! You beat me... again!! And I just spent my entire savings on these new Koopa Mach 1 Sprint shoes!" - Koopa the Quick (Mario 64)