Make sure that you get a cleaning rod that will fit to .22 The one I have (brand is Kleen Bore) measures .202" diameter and has the 8-32 thread, so it just so does work in a .22. Some rods are "fatter" - won't fit. The Tipton jag set mentioned above has the .22 jag that you want, and a .17 - but the .17 is smaller thread - had to get an even skinnier cleaning rod for that one.
Also, I could not find or make patches thin enough to go into .223" bore with that Tipton jag, so had to peel it down a bit on the drill press, so it would work with what I have for patches.