Big plus on the Otis system. I've been using them for years on every gun I own. I buy mine on eBay for cents on the dollars. Just know what you are buying and don't get into a bidding war. Use esnipe. You can use any brushes, tips or patches. The patches, just cut a slit on your own.If you only plan on having a .22 rifle, then any decent rod, some jags and patches, and a bore brush along with a bottle of Hoppes 9 and some Rem oil will do you for a long time.
If on the other hand you end up like many of us do, with multiple caliber handguns and rifles, a decent cleaning kit would be less expensive in the long run.
I have recently been mostly using the Otis system and it does fine. A flexible coated cable that you pull through from the muzzle end and various size brushes, etc. It all fits in a little pouch. There are various size kits, but this one from Midway is a good starter.
Otis Deluxe Rifle Pistol Cleaning System