Hoopy, there are several things you need to consider. Your rifle has enough power to kill a bear at 350-400 yards, but you have to be sure of hitting the bear in the vitals to make a clean kill. If you just hit the bear in a random spot you will have one really pissed off bundle of fur looking to hurt somebody. You need to put a high quality scope on your rifle to help you put the bullet in the right spot. I suggest Leupold or Nikon. Not the cheapest but you will not regret having a good scope that will let you have a good clear view when the light is poor outside.
Also, it is one thing to bounce around a pepsi bottle. It is something else to try to kill an animal that can bite back. You will need to practice with your new rifle under real field conditions. Get off of the bench rest and shoot off of an improvised rest or from shooting sticks. Practice the way you will hunt.
Another thing to think about is your choice of ammunition. Winchester makes fine ammo, but for bears you need to use a bullet that will penetrate deeply and not tear itself up on the first bone it hits. Do not use the Winchester bullets with the plastic tips, they are designed for thin-skinned animals like deer. Instead use the Winchester Fail-Safe bullets, they are built for tougher animals. As an alternative, look for ammo loaded with Nosler Partition bullets. They are excellent on tough game.
Finally, to really improve your odds of sucessfully taking a bear, try to get within 200 yards before you shoot. The bullet will have a lot more "punch" at closer range, and you will still be far enough away from the bear for safety. But have your brother-in-law with you for backup. He can cover your rear if the bear does happen to run in your direction after you shoot, and he can help drag it to the truck after you make a clean kill! Take a camera to record the stunned look on his face. Good luck