I had a carpeted stair to the garage like yours and it just got dirty and torn up from overuse. But my garage was a workshop and generated a lot of dust. I ended up taking the carpet up and having a bare step. My intent was to cover it or rebuild it with composite decking, but I sold the house before I got to it.

The tools you need for regular carpet laying are a carpet knife, stapler, maybe one-sided carpet tape to seam the pieces together, a knee kicker to stretch it. But covering the step would be more like upholstering. So a stapler and a box cutter might do, but it depends how you would wrap the step. If the carpet is cut pile, you might be able to tuck the edges. I would try to avoid taping the wood step because the glue would be a pain to remove. A manual stapler may be difficult to use on that step - for my stair runner, I bought a pneumatic stapler to use with my compressor. Cover the top of the step with one piece or two seamed together, then wrap the bottom with another piece if you want to (or stain or paint the bottom if you prefer).

I would do a different treatment for the step (not carpet). Then store the extra carpet in a climate controlled area in case you need it to replace a section inside the house. If it is solid-ish cut pile, a piece can be replaced with carpet tape relatively seamlessly. This is useful if you have pets or a spill, or other damage to a small section of carpet inside. Those spare remnants of carpet could save you a full carpet replacement in the future.

Check Harbor Freight for tools - sometimes they are a lot less expensive than other stores.

Thank you @kelli_ga. I'll consider another treatment. There's enough in the roll to do the step and have a square yard left over for patching. I've got a few different utility knives/box cutters and staple guns. I don't have any pneumatic tools. Would an electric staple gun have sufficient power? eg, https://www.lowes.com/pd/Arrow-T50DCD-3-8-in-Cordless-Electric-Staple-Gun/1000224145 ?

I say paint it and apply some non-skid strips, there are some good self stick ones. An outdoor rug on the floor to get crud off of shoes you can hose off.

It's going to become a door mat and look awful in short order.