You might have success covering the patch with cardboard or several layers of newsprint, then covering that with a thick layer of mulch. Pulling out by hand is too much work. Let the worms eat the Johnson grass for you!
Depends on the scope. Sheet mulching works great if they are in small colonies. Larger, might be hard to get enough material to sheet with. Covering for 2 growing seasons if you are reusing the sheeting.
Pulling smaller plants that haven't spread, or seedlings is probably pretty efficient. Learning to identify the grass from the ligule might be a good idea. https://en.wikipedia.org/wiki/Ligule
Repeated and close mowing kills Johnson grass seedlings, prevents seed production, and reduces rhizome growth and regrowth of shoots.