Remove the screws from the metal handrail mounts on the underside of the wooden part of the railing using a screw gun. Set the screws aside in a safe place; you need them when you replace the wooden handrail.
Lift the wooden handrail off metal handrail brackets, and set it aside. Be careful not to damage the wall or surrounding woodwork when you are removing the handrail from the stairs.
Place the replacement wooden handrail on a work surface in a well-ventilated space. Brush an even coat of wood finish onto the portion of the wood that is facing up. Allow the finish to dry for at least four hours. Rotate the handrail so that the unpainted half is facing up, and complete the first coat. Allow the second half of the first coat dry for four hours, and apply a second coat in the same manner; be careful to avoid runs or uneven coverage. Wait 24 hours after the final coat before continuing with the project.
Center the replacement wooden handrail on top of the metal handrail mounts. Drive the screws that you removed in Step 1 through the metal handrail mounts into the bottom of the wooden handrail using a screw gun.