To get the sticker off -- carefully peel what you can off. Lightly tap the sticky side of the label or scotch tape to the sticky that is left on the cutter to "lift" the remaining sticky. If there still is some sticky left rub it with a bit of peanut butter (creamy) and wipe with a paper towel.
Can't guarantee it will work all the time, but it will most of the time.
Good luck and thanks for the tut. Great quick gift idea.