Is there a lot of rust in the bore? If not, just use a bronze brush with Hoppe's solvent, if you can get it through the bore. I've rarely plugged a bore unless the bore is full of rust, then used Hoppe's, or Break-Free, letting it sit in the bore overnight, then flushing it out and trying to get a rod through it. The key is to remove any rust gently.
If it's not rusty and you can get a bronze brush through it, just use bore cleaner and brush it, then use patches to check the progress. I've cleaned a lot of .22LR bores and found that the wax on bullets prevents rust to a great extent. In fact, I've NEVER encountered a rusty .22LR bore! You can do more damage to a .22LR bore by being too aggressive in cleaning than by shooting.
Tread lightly and save the "plug" as a last resort.