The spinning barrel at the end can be solved by jumping on it and holding Up or Down to match the direction of the barrel. It will make the bounce distance enormous enough to hop off on the down swing.
For example, you jump on the barrel and it naturally goes down, so you hold down. It will go down 20 pixels. As it starts to go back up, you hold up (no need to jump or anything) and it will go up 40 pixels. On the way down you hold down and it will go down 80 pixels. Repeat until the bounce goes so far down you hop off below with ease.
The numbers are made up, but the process is correct.
I imagine as kids we eventually got through it on accident. Like out of frustration we mashed the down button to make our jump land harder and it actually worked xD