There's not much of a pattern to write, because everything depends on
the form of your mug - definitely not a beginner's project, could barely
cope with it myself, but the colors do make up for other shortcomings
So these are just some basic guidelines!!
The idea is: making a foundation chain for the bottom circumference, and
then doing sc's (or dc's, or trebles) round after round - but you must
do increases in nearly every row or just occasionally, depending on how
your mug is shaped. Constant fitting is crucial.
Also, after the foundation chain, in the second row, you must leave an
approximately 2 to 4-ch space (free for the handle) and turn in every
round so as to avoid that space.
At the end, you may crochet a border all round the thing, add a button
and some chains to the other side to secure it. Again, length depends on
how tight you want your buttoned-up cozy to be.