Speaking from experience, there are no cat breeds that are more likely to catch mice. However, you can encourage this sort of behavior by training them to catch wild critters! Buy toys that are supposed to be chased, don't overfeed them, encourage this sort of behavior by praising them when they do start taking home their catches. Another thing that I found, based from experience, is that tomcats are more likely to give chase after mice and other little critters. But that might just be coincidence on my part! About the age of the cat, I think it's better to have them as kittens and then encourage this sort of hunting behavior that way.
