Is there a way to print on a 3D printed surface?
Rather than post a series of links for various services and vendors, consider to use as a search term "hydrographic film at home" to find results to suit your purpose. The first search results I found excluded "at home" and indicated it was not suited for home/hobbyist use, but the modification resulted in vendors supportive of home users.

In advanced circumstances, the image printed on the film is topographically matched to the model to which it is to be applied. For home users, I suspect that is somewhat out of reach. Most of the products aimed at the home user are patterns or solid colors, although with care, one could combine films in sequential applications.

Effectively, the process involves a colored film floating on a water filled container. Certain products are applied to the film and to the prepared model to ensure bonding. Additional processes are required to protect the graphic-coated model after removal from the bath/container.

As with the other responses, it's not likely one will achieve 300 dpi resolution without resorting to the topographically matched commercial methods.