Remove the chair cushion if possible and clean out the area where the arm fabric meets the seat. Use the knife to cut a 6-inch slit just above the crease where the arm fabric and seat fabric meet. Widen the slit slightly using your fingers to allow easy access for the stuffing.
Fluff the cotton batting by working through it and stretching it out, to give it bounce and occupy more space. Begin to stuff it into the arm of the chair using the dowel. Stuff the arm evenly by using the dowel to spread it throughout the arm. Continue to stuff the chair until the arm is firm to the touch, putting in extra stuffing to compensate for the eventual softening of the arm.
Stretch the fabric above the slit down over the slit and staple it in place using the carpentry stapler, making sure that the fabric is pulled over a section of the wood chair frame. Repeat for the other arm.