Re: sniper rifle under 1000
Look for a Rem 700 SPS or Savage 10. I'd suggest ditching the stock immediately and look into a HS precision, B&C, or Choate stock, and decent lower level optic (SWFA, Bushnell, Weaver, etc). Those 3 would be right around your $1000 mark. Do a quick search here on other people's budget builds. That will give you some visuals on lower dollar builds guys here have done, and maybe some ideas for yourself.
Word to the wise (since you mentioned ammo $), for plinking or practicing- cheap ammo is ok, however if you get serious, cheap ammo is... cheap. Many of us reload for two main reasons. Quality and $. Quality meaning you can tune a load for your gun that will *usually* work better than factory match ammo. $ meaning after initial investment of press and dies, your price per round drops significantly (last I checked I think my .308 $/round was somewhere around 47 cents)
Hope some of this helped. Good luck and Welcome to the Hide