Remove crumbling plaster and peeling paint, using a putty knife.
Apply a mesh patch to the hole. Plaster patching kits have the proper metal or fiberglass mesh. Lay the mesh across the hole and follow the instructions on the kit to secure it to the plaster surface. Cracks in the plaster do not need mesh covering.
Smooth joint compound over the gap with a utility knife.
Cut down fiberglass mesh tape to the length of the hole or crack, using scissors. Press the tape against the wet compound until it sticks. Smooth the compound over the surface of the tape to even out the area. Let the compound dry, following the recommendations provided with the product. Usually, this will be at least overnight.
Add two more coats of joint compound to the damaged area. Pull the edges of the compound past the margins of the damaged space to blend it to the ceiling surface. Allow the compound to dry after each application.
Sand the surface area of the patch with a sanding sponge to smooth the surface.
Prime and paint the ceiling, using primer, latex paint and paintbrushes or rollers.