To be more concise:
Call 1-877-636-7426
I was asked for the following information:
-The first and last name of the card holder
-The complete billing address
-The reason for cancellation
That is all. The representative told me how long my premium service would last (how many days until the end of this billing cycle), and that my card would not be charged again.
Now I will admit to having some difficulty finding the phone number at first, but it is ALL over your premium account. It can be found at the bottom of the Neopets Portal, AND at the bottom of the Neomail email service. Now if you don't remember the billing address- then I HOPE you saved the welcome email that was sent to the neomail account when you first signed up. That has all the information you need.