A bull barrel is another way of saying an extra heavy barrel.....used primarily for bench rest target or long range shooting.
Whether a rifle has a short action or a long action depends on the caliber. Short cartridges (.223, .308, etc.) will have a short action. Long cartridges (.30-06, .270, etc.) require a long action.
Take a look at the Savage 12VFSS: http://www.savagearms.com/centerfire/varmint/12fvss.htm
Sounds like what you're looking for and you can probably find it discounted for about what you want to spend