imo scope is as important as the rifle. for 400 yd shots id stay with the "VARMINT" calibers. im a big fan of the 243. you can get a heavy enough bullet to buck the wind pretty well, recoil is very little, and when everyone goes ar nuts 243 ammo is easier to find than 223. but to be honest either would acomplish what you want. 243 ammo is a little more expensive.
as far as rifle for 350 id try and find a stevens or a really good deal on a used accutrigger savage.
I have not really ever been interested in bolt action rifles but I am getting one in the next few months to plink with, I have a limit of 400 yards where I normally shoot. I want a decently priced and readily available round. I wont spend more than $350 on the rifle. What would you suggest?