For one mug you would go to a color copier that has been set up with the proper dye sublimation cartridges. A good working knowledge of four color printing, and control of curves are a must. Encad plotters with dye sub cartridges work fairly well. The speed of the printer fits the job. The downside is the spot size.
Darker images seem to hide the spot.
Here, however, you may run into a quality issue. Screen printing will not give the resolution that other processes can give. Flexo is best suited for specialty printing, such as shoe laces and name tags. The substrate to be sublimated and the quantity will dictate the equipment. Specialty presses to do caps, mugs, and plates are available.