Scrape off peeling and/or damaged bits of paint with a putty knife. Scrape the surrounding area to remove loose bits of paint. Scrape underneath any peeling paint to ensure the entire damaged section is completely removed.
Wipe down the damaged section with a damp cloth.
Brush on primer-sealer on the damaged section with a paintbrush. Let it dry for 12 hours.
Spread on joint compound onto the damaged section with a joint knife, buttering the surface with 1/4-inch of compound. Skim off excess joint compound while filling in any depressions, cavities, and holes to ensure a smooth, even surface. Apply several thin coats rather than one thick coat, letting each layer dry for 30 minutes before applying the next. Skim on the surrounding areas to blend in.
Sand down any rough patches with medium-coarse sandpaper in between coats and wipe down with a damp cloth.
Skim on a final coat of joint compound with a joint knife. Apply perpendicular to the previous coat. For example, if previously applied with a horizontal stroke, apply the skim coat horizontally. Let it dry for an hour.
Sand the repaired area with 120-grit sandpaper. Wipe down with a damp cloth.
Prime and paint as needed.