Locate and disconnect the spark plug wire boot from the tip of the spark plug. This is typically located on the front of the mower. It is a red or black rubber insulated wire. Pull out firmly to disconnect it, preventing the mower from starting accidentally during your repair.
Use a nut driver to remove the screws from the metal, or plastic cover on the top of your lawnmower. These are typically located at the bottom of the brackets holding the cover up above the engine. Turn the screws counterclockwise to loosen and remove them. Lift the cover from the mower. Use a knife to cut the rope, holding the pull handle onto its end, and thread the rope through any brackets on the mower handle to loosen it.
Turn the cover over and set it on a sturdy work surface. Use a socket wrench to loosen and remove the nut in the center of the plastic wheel inside the cover. Pull the wheel free and pull the old rope with it. Position the new wheel onto the spindle in the top of the cover, positioned as the original wheel was. Thread the nut onto the spindle and tighten it with the socket wrench.
Remove the plastic band, or wedge that holds the spring steady as part of the packing in the new part. Keep your hands well back, as the pulley may spin rapidly. Pull the rope out from the pulley and thread it through the hole in the cover. Replace the cover on the mower and install the mounting screws.
Thread the rope up through any brackets on the mower handle, and feed the rope through the hole in the handle (if the rope was extended to the top of the handle before). Tie a knot to hold the rope on the handle. Reconnect the spark plug wire.