A lot of CZ shooters do use a smaller diameter rod such as a .20 cal. but a .22 cal Tipton fiberglass rod will work without issue. That's what I used for years on my CZ rifles. It was a good rod until this year when it broke where the shaft and handle meet.
Unfortunately for the OP, with his self-imposed budget of $30 - $40 for a rod and accessories such as jags, oil or solvent, and patches he will find that his choices are somewhat limited. Segmented rods -- the three piece kind that screw together -- should be avoided if possible because they are weak, especially at the joints. One piece rods are much better and it is possible to find an inexpensive one piece rod within budget. The satisfactory Tipton fiberglass rod and the better quality stainless steel rods would take up the entire budget, perhaps more.
No rifle should cleaned from the muzzle. Always clean from the breech, even if a pull-through is the only option. When using a cleaning rod, a proper bore guide is a must to ensure that the rod doesn't damage the chamber.
