Lightly sand off the peeling paint with sandpaper. Sand the rest of piece with a light hand, just enough to dull the finish and roughen the surface. A slightly roughened surface creates a better bond with paint. Fill in any small chips or gouges with wood filler.
Wipe away all sand particles with a tack cloth.
Coat the pressboard with one coat of quality latex primer. The primer helps to seal the material, blocks stains and prevents the original color from bleeding through the new coat of paint. Primer also helps the paint to stick to the material. Allow the primer to dry as recommended by the manufacturer. Apply a second coat of primer if the new paint is much lighter in color than the old finish.
Paint the pressboard with a quality latex paint. One coat is usually sufficient but you can apply a second coat after the first if desired.