In Extra Mode, the rope is no longer in a barrel, instead it's in a picture of a white cat nibbling on it. The scrap of paper no longer tells you where the rope is, nor does it tell you how to get it. To get the rope, you need to give the Yarn Ball to the white cat, the rope will drop down from the painting to Viola.
The rope is meant to be given to the Librarian, upon giving the rope, he will thank you by giving you the Book of Death. You can choose not to give it to him at the start, and he'll still give you the book when you decide to give him the rope, although he'll be less enthusiastic.
When you return to the Library after freeing the butterfly, you will find that the Librarian has hung himself with some rope, most likely the same one you gave him.