Savage action would be the budget choice. Do you reload? Are you willing to wildcat? Is this for some type of competition?
I'd take a 6.5mm or 7mm over a 30 caliber in most cases. 260 Remington, 6.5 Creedmore, 6.5x284 would be excellent choices, in that order. Even the 7m-08 is a very good choice if you want to stay out of the magnums. All will best a 308 out to 1000 yards.
500 yds+.... and im talking about a full build. ive heard great things bout .300 win mag, but ive hear even more bout .338 lapua. i do not reload but a buddy of mine does and he will do it for me until i get everything i need. also i know that ya need to spend atleast what you have in the gun on the optics. so im sayin $1000 for just rifle.
If you snagged a Savage 110 long action for $250, and got a Criterion barrel & NSS recoil lug for $350, you'd have $400 left for a stock/trigger/bottom metal/magazines in order to hit your $1k budget.
Yes you can build a Savage for under $1k...just depends on what you want, what your expectations are, and your interest in eyeing multiple forum classifieds for deals on used gear.
Also, you don't need a magnum for 500yd+; a 260 or 6.5 Creedmoor will easily make 1000yd and beyond.
for 500 yards I would not consider the 300 winny, or the 338 lapua. I would look at the 260, 243 area of calibers. Since you dont reload the 308 has good factory ammo options so you might look into that.
So you need a short action and I would go with a savage, so
You can buy a Savage 11 short action package gun or a Stevens 200 for $350 or so. Sell the factory barrel and stock, and let's say you've got $300 in an action.
Then call Jim Briggs @ Northland Shooters Supply and buy a Criterion barrel in the caliber of your choice (I'd recommend 6.5 Creedmoor if you don't reload yet) and NSS recoil lug...roughly $350 to your door.
Then buy a Bell & Carlson A2 Medalist stock from Red Hawk Rifles...roughly $250 to your door.
That puts you at $900...but you'll need a barrel nut wrench and a go-gauge for the barrel swap which will put you pretty near $1000.
Your not going to get this for your budget but here is my stiller 260
Your not going to do this for 1000 either but you will be closer than the first rifle. It is a model 70 358 bfg I did. Since the pics I threaded the 358 for a VIAS brake