With a brush, you have no choice other than from the muzzle to the back. However, with a bore snake you can bring it through the barrel from the back to the muzzle.
If you feel the need to remove some carbon/wax buildup, I would suggest a Patchworm (or make your own out of trim line). You pull a patch with solvent or oil from the forcing cone end.
I don’t think I’ve ever had the need to use a brush on a .22.
Alternately, use wipeout and pull a patch through.
Unless there is a specific need, I would not bother with a rod and brush on a .22 barrel.
ETA: I once had a smith 617 that I stupidly sold. I do seem to recall pulling a brush with some big 45 wrapped around it to clean lead from the forcing cone before I sold it. You can insert your rod, screw in the brush and pull it through. Just another option.
Otis has a small caliber (to include 22) cleaning kit that has a pull cable with attachments for a brush and patch holder that allows the back to muzzle direction. I prefer patches to see how clean I'm getting things but overall I think a bore snake does a better job than this kit.
For my rimfire pistols I use weed eater string and a proper size patch and solvent. It gets the job done nicely and when it gets gritty cut you off another piece. No fuss no muss