I have several 22's that are tack drivers and don't even own a cleaning rod. All I ever use is a patch worm.
While you are collecting ammo to try, get a box or two of Wolf Match Target or SK std +. You will have better luck with subsonic ammo if you are going for accuracy.
By the way, if you find one of those 22 rifles that produce ten shot, quarter inch groups at 100yds, better buy two of them, they're rare.