Turn on your hunter's sense and try standing back from the direction he hits you from first. You will see him come down from the mountain. Make sure you have your best revolver out. Shoot when he comes out of the bush, Keep your hunter's sense on the entire time.
Use the the .416 rifle. When you first enter the "arena", move to the left. There should be a cave. Wait in front of it for the lion to charge. Right before it hits you, shoot it. Then immediately jump into the cave. Go out the other end. Move straight for the lion. When you are close enough, it will turn around and swing at you. Shoot just as soon as it turns around. Note: You must keep running at the lion before and after you shoot it. By doing this, after you shoot it you will be close enough to the cave to jump back into it. Do this repeatedly until the lion is dead.