Paint or marker for insulated cup decoration
You can fire artwork on porcelain and it will be permanent.  Pretty much any other surface won't be.  Even commercially applied artwork wears off and washes off eventually.
If you know the actual surface material/finish (like by contacting the manufacturer), you can optimize the decoration for adhesion and durability.
You can protect the decoration by building up a thick coating (via many light coatings) of a clear finish of some kind over the decoration.  A protective coating also makes optimizing the decoration material less important.  I'm not familiar with that particular cup but if the protected surface extends to the lip you drink from, the coating should be food-safe.
If the coating starts to wear off or gets scratched or chipped, you could reapply it or touch it up while the decoration is still intact.