Do sugar cubes disintegrate when being painted?

No

There are edible paints used by specialty bakeries that can be used on cakes etc. With a light touch and an airbrush used specifically for edible paint, you can paint on sugar cubes.

Also, if you are just using the cubes as a base, with no intent to eat them, you can in fact spray paint them.

I would however suggest a good airbrush and not an aerosol spray can.