Rodents. What type? Mice? Rats? Prairie dogs? Beavers? Cats can help with some but not all of those.
Typically a "rodent problem" refers to mice, and in particular, mice in a building. Most don't care about mice out in the fields. A cat can sure help with this but not all will. Some are too lazy to make good mousers. We just gave a darn good barn cat to someone who needs this type of help, and hopefully the cat will acclimatize to that barn and get to work.
On the other hand, I've heard that cats won't kill bigger animals like rats, unless they really need to, preferring to live and let live as long as there is enough food to go around, but again, this will depend on the cat. I used to have one that regularly brought down full adult rabbits. But I also had one that refused to hunt, wanted to be fed, and simply could not cut it as a barn cat.
I think you have to look at the type of rodent you are trying to control, is this in a building or out in a field, and then be careful about the type of cat you introduce. I think (or at least I hope) it goes without saying that whatever cat you get should be spayed or neutered. There are usually local clinics and plans to have it done in a very cost-effective way. You could check at a shelter because sometimes there are basically feral cats that are breeding out of control, so they can be trapped and altered, and then taken to your place. Such a cat would probably do a good job at hunting.