Flattening out the lip in a two dimensional image won't do much for you in this case. It would be much like trying to flatten out a hat.
Since the tapered lip is essentially a cross section of a cone, it's possible and reasonable to flatten out a cone. Here's the brute force method: Make a paper cone that perfectly fits over the lip. Mark the height of the lip on the bottom of the cone. Now flatten the piece of paper from which you fashioned the cone. The strip at the bottom, which should be an arc, where you marked the height of the lip represents the exact size and shape of the piece of vinyl you'll need to wrap this section of the cooler.
You can also do this analytically. That would be with conic geometry and no physical models. The necessary formulas are readily available on the web. This method requires at a minimum the diameter of the base of the lip, the diameter of the top of the lip, and the height of the lip. It may prove more difficult and thus inaccurate to take these measurements than to simply create a physical model.