The over-knitting is what I would likely do. Basically, pick up stiitches from the sleeve, and start knitting back and forth until your 'patch' is longer than the damaged area, then decide how you want to attach it to the sleeve: stitch it down, or bind-off it down. Stitching is easier/faster.
There are a lot of details (like make the patch bigger than area to which it is attached) so ask a lot more questions!
And do not be afraid to stop and do it over. You can always improve it!