I'm not too sure if there is much you can do to get your gun up faster, other than practice. I do suggest that you get it all the way up to your eye before you shoot though, even if its a longer shot. You have a better percentage while aiming the gun at a farther target than you will shooting a closer target without aiming. To better your quickness of bringing a gun up (I have no idea if this will work or not, but you can try), whenever something goes by, or jumps up ( "flushes" ), try shouldering an imaginary gun. Like if you are on a walk and a bird jumps up and flys, "aim" at it. I hope that can help.
The life cycle of a hunter... hunt, eat, sleep if there is time, then go hunt again.
Chuck Norris doesn't go hunting, becasue hunting infers the possibility of failure. Rather, Chuck Norris goes killing.