BTW, this will never happen with the strong laminated product we use. The cats will bounce right off!
We perform repairs on shoji screens if possible and feasible. Most repairs involve replacing the broken "rice paper". Note that we use a strong modern laminated material that requires to replace the old paper in the whole screen, not just a single square in the panel. Also, most older plastics/papers have discolored over time. They have become yellow and brittle. If the material in one screen needs to be replaced, it typically cannot be color matched with the other panels. In other words, to make all screens look uniform, you may have to replace the material in the other screens in addition. Some screens are very difficult to open to access the material; those may not be worthwhile repairing.
Given the variability of issues, we cannot give you an exact estimate without looking at the broken screen first. Please call us to discuss the situation. Mahalo!
Notice how the old paper has yellowed (it was white, once). In this repair, which was in a rental unit, we used a very strong replacement material: .45 mm Cloud Dragon paper.
After 30 years of abandonment in a pocket, these 3 screens have been restored to their former glory.