If the cleaning rod is good, all you need is a brass brush and patch holder and some cleaning patches (that should each screw to the end of the rod), and some bore solvent. If the original cleaning rod is unthreaded for cleaning tools, you will need a modern cleaning rod that is (one piece is better than multi pieces). Get one long enough for the barrel, but not super long as this is like working with a fishing pole.
Remove the bolt (from the rear), and clean from the breach (not from the barrel).
The Carcano was 6.5mm, so you need a brush for .26 caliber. The picture is a full size/length rifle, Oswald used the shorter carbine Carcano (I think).

Bore cleaner CLP (clean, lube, preserve) is good for clean and oil of whole rifle.
https://www.amazon.com/BreakFree-CLP4-BF-CLP4-CLP-Liquid/dp/B0002IKDQ6Straight bolts are not as user friendly as curved ones.
Before you shoot with modern (or any) ammo, you may want to have headspace checked by a smith. Excessive headspace (how a cartridge seats in the breach of the barrel) can be dangerous in Militray Surples rifles.
https://criterionbarrels.com/media/what-is-headspace-and-why-should-it-be-checked/?v=7516fd43adaaOr, sandbag the rifle on a bench and shoot with a string to the trigger and if nothing blows up, look carefully at the spent brass for any case bulges near the base/rim of the cartridge. Damaged case heads from spent rounds means bad (excessive/eroded) headspace. Still wear eye protection.
If you call Midway USA, you can get to talk to someone who will know what you need.
https://www.midwayusa.com/brushes-swabs-and-mops/br?cid=8873https://www.thefirearmblog.com/blog/2020/03/06/65x52mm-carcano-steinel-ammunition/Be careful of surplus ammo, it's probably ancient. You can get a hang or squib fire where the bullet hangs up in the barrel. If you shoot another round behind it, the barrel will probably rupture and explode. A hangfire bullet must be punched out of the barrel with the cleaning rod (and screw up the threads).