I would wrap a line of masking tape around the edge of the cup on the outside at the top, because this modgepodge gloss appears not to be food safe. I would also wrap at the bottom for aesthetic reasons.
Although it is water soluble, do not dilute it.
Put the mug to be painted upside down on stick it on a plastic cup. Use the biggest, softest synthetic brush you can find and quickly brush downward in an even stroke from tape to tape. With the next stroke, overlap by 1/16” or 2mm and work your way around. Depending on the viscosity you may need more or less paint.

Personally I would dip the entire outside of the mug in polyester resin. That water-based stuff probably won’t hold up to abrasion and if it isn’t food safe I wouldn’t want to accidentally drink a chip. Something like this would work. Use an S-hook to hold the handle, dip it in the mixed resin, hang above the dipping container for five minutes and then place on a coaster to cure. Use in a well ventilated area and don’t get it on your skin or in your eyes.