Your question is about of particular implementation of Anamorphosis.
HELPFUL ADDITION FROM DISCUSSION:
"You should stretch the picture horizontally. I saw some guy sketching the circle on the mug - until it looks perfect to him, then placing on the mug semitransparent paper and copying the "circle" (oval actually) to the paper. Then scanning it - the resulting image match perfectly to the needs - you should understand that he scaled the original pic to the scanned oval."
In order to compensate the spatial image deformation on uneven (i.e. round surface) you should know the observer's point of view - because the picture cannot be seen as "normal" from multiple points you should choose one and the most frequent point is just central front, similar to the situations like when you normally look at the pictures in the museum (you should visit museums regularly!).
For task implementing print on curved surface you should use node deformation of the source image REVERSED to this image -
You cannot create undistorted image wrapping the whole mug, because this implies multiple points of view. I can advise you to print smaller image which not wrapping the whole mug.
I should state that I created anamorphic images by hand and by laser projecting (you can google the topics) and it is absolutely fun: I take a shape and placed it on some distance; next from the observer's point of view I projected laser beam on the borders of the shape (it was just text label as I saw in the internet) and by marking the laser beam projections on the walls I created anamorphic text, similar to this -