The only think I can think of in your price range to do what you want to do is a muzzloader. You can find them used. You can turn a muzzloader into a shotgun and you can shoot 200 yards with a muzzloader using good bullets and calibrating the charge. Muzzloaders are also quieter than rifles.

If you don't like the muzzloader idea then I'd save pennies till you have about $500. Don't forget about the cost of ammo. The ammo may cost you more than the gun in short order.

I like the suggestion of a .243 win. Good cartridge.

Tom