Lay out the linoleum on the treads and risers to find the best placement for the tiles, and determine where they will be trimmed to fit. Begin at the top of the stairs; if the top step is finished with the flooring on the rest of the upper level, start with the top riser; otherwise, start with the top tread.
Place the first, full linoleum in the center of each riser or tread, and lay the next pieces evenly out to each side. This buries the cut pieces in the corners of each step where they are less noticeable, giving a balanced installation.
Place the rubber nonskid edging on the top, front of each tread at the same time you lay out the linoleum. Trim the linoleum so it covers the metal profile of the edging, but sits flush to the rubber portion. Trim each tile for the riser so it comes up to just below the front of each nonskid edge.
Cut the linoleum to fit each step with a utility knife. Set the linoleum down on a flat surface and place the tip of the knife against the tile. Pull the knife firmly across the linoleum to make the cut. Test each piece for fit on the step before you install it.
Install the linoleum tiles from the top of the stairs down. Spread the latex adhesive onto the steps with a trowel, and press each tile firmly into place. Set the metal profile for each edge first, then the linoleum behind it. Set the rubber into the profile last. Install each tread then riser until you reach the bottom of the steps.
Roll the linoleum on the stairs with a rolling pin to help the linoleum bond with the adhesive. Apply as much pressure to the pin as you are able, moving it several times across each tread and riser until you reach the bottom. Refrain from using the stairs if possible for the next 24 hours to give the adhesive time to cure.