22-250 for 350 yards? NO.

You can use anything from a 243 up and drop a deer dead on the spot if hit right. I personally don't like neck shots seen two deer lost in to years from bad neck shoots. I would look at a 30-06 or a 300 wsm personaly and hit them with a high shoulder shot its a little bigger target then the neck.

Also if you ''hope'' you hit something you shouldn't have pulled the trigger. Its not hard to break the spine and drop a deer. hit the high shoulder and the shoulder blade bounces off the spine and the deer usually drops, usually. Nothing is definate no matter what gun or round you use.