You have to coat the rusted area fairly well until the rust turns a dull dark grey. Rinse and dry well and paint as soon as possible. To bring out the color of the original paint I would go with a heavily thinned clear coat.

Will the phosphoric acid take care of the rust without harming the painted graphics below?

You say rinse and dry well and paint as soon as possible.

Do you mean - rinse and dry both the rusted area and the paint beneath? Or - Are you saying after rinsing the rust off after the application of the acid - that I should paint the cabinet? Not sure I'm understanding what you mean.

When you wash it off you have to wash the entire piece. If you're not repainting the rusted areas with color you might be able to just clear coat the rusted areas but if you repaint the entire piece with clear coat it will brighten the old finish.