Another vote for a bolt action:
.243 Winchester with 105gr (A-Max) or 107gr (Sierra MK) readily handles ranges to 1K. Standard, readily available, low cost components in a proven caliber.
Start with a Savage Accutrigger short action from the cheapest rifle you can find, new or used. Add a 24" Shaw stainless varmint weight barrel with the 1 in 8 twist necessary to spin those long bullets for $200 plus fitting. Drop into a Choate Ultimate Varmint or Tactical stock for another $200. Add your choice of mount and scope and go shooting.
You get a roughly 12 pound gun that is well adapted for long range, has very mild recoil, and avoids odd-ball or proprietary components for a very reasonable cost. If you bought the barrel sleeve wrench, you can swap out barrels to try other calibers for the cost of the barrel.
IMHO