A "quick & dirty" solution for the brass driving plug
After thinking about it, this could be a better solution than the countersink-head method!
If you haven't got a lathe, and can't find a suitable countersink-head brass screw, try this:
Get hold of a 1/4-28 brass screw. Saw off the head, to leave you with a threaded stub - 1" is quite long enough. File the thread right down to the core at both ends, to leave about 2 turns of thread in the middle (minimum: 1 turn with overlap). The core diameter of 1/-28 should be a sliding fit in a .22 bore. Check that it really is so, and file to suit. You now have a very simple driving plug, which can be driven from both ends, the remaining 1-1/2 to 2 turns of thread serving as the driving band.
Oil the bore, hold the rifle/barrel vertically, muzzle down, and drop the plug into the chamber. Use a good (yard?) length of 5mm or 3/16" brass rod to drive the plug down the barrel. The initial resistance will be quite stiff, until the thread has been formed to fit the rifling. If the obstruction is dried crud, this will shift it. If not, you can drive the plug back out from the other end.