Mount a tack-less strip across the landing about 2 inches short of the spot you expect the carpet runner to end. Use a hand saw to cut the strip to the same width as the runner then use a hammer and nails to fasten the strip to the floor. The tack-less strip should be facing away from the staircase.
Install carpet padding onto the landing. Use scissors to cut the padding so it wraps around the front lip of the landing and reaches up to but not onto the tack-less strip you installed at the other end. Staple the padding to the underside of the bull nose. One staple every 3 inches or so will do. Fasten the padding to the landing surface by driving staples every 3 inches around the entire perimeter. Drive staples every 6 inches or so throughout the middle of the padding.
Lay down the carpet runner with the grain facing toward the front of the stairs. Set one end of the carpet onto the tack-less strip and press down until it grabs on firmly. Allow an inch or so of carpet to extend past the tack-less strip on the opposite side. Apply tension to the carpet and pull it toward the front of the steps. Wrap the front edge of the carpet over the lip of the landing and use the stapler to fasten it to the underside of the bull nose. Use a carpet knife to cut away any excess.
Install 1/2-inch carpet tacks along both edges of the landing runner to prevent it from lifting. Place each tack as close to the carpet edge as possible and use the hammer to drive one tack every 6 inches or so.
Mount a metal saddle over the exposed end of the carpet. The saddle should cover the carpet seam and abut against the tack-less strip without overlapping it. Use a power drill to make pilot holes into the floor then use a screw gun and the screws provided with the saddle to fasten it in place.