I have a friend here in Canada that got into Biathlon recently. He is a member of a local club that has their own rifles, which he gets to "check out" in some fashion, so he can practice on his own time. This might be a good option for you, if available where you are.
I'm no gun expert, but from what I know, that last little fraction of a percentage of accuracy is what you need in high-accuracy shooting... and that's what costs the most. You might be able to find a used biathlon gun somewhere, but I doubt you'd be able to find a hyper-accurate gun for $500.
One place you might check out is the "Rimfire Central" forum. As the title indicates, it's dedicated to only rimfire firearms, and there are people there into all aspects of rimfire shooting.