I reckon all your replies are correct so far. The opossom has a brain case about 1/4 to 1/5 of a similar sized coon. Possums are marsupials so their brain structure is a little different than that of mammals. Small brain cases and small brain structures make for a small target. Human suicides that put a bullet to their temple can and have survived. They either severed their optic nerves blinding themselves or destroyed part of their cerebral cortex resulting in altered personalities and diminished life quality. A bullet through the brain stem is the only way to shut a mammallian brain down instantly. A brain stem hit will halt all cardiac and respritory function. How big is the brain stem of a possum. Pretty darn small...I'm guessing about the same caliber as a 22. I don't think I'd be able to hit that sweet spot at a few paces but I imagine a few rounds of 22 through the brain case will eventually damage the brain stem enough to do the trick.
Then again there is that primitive reptillian brain thing. I watched my Mom's cousin butcher three snappers when I was a kid. They were hung upside on a tree over night without their heads. The next day he cracked one open, cut out the heart and plopped it in may hand...the darn thing kept beating for about 5 minutes. I'm guessing the the spinal cord has enough neurons to keep part of the critter functioning. Possums are primitive critters and may share some similarities with the reptillian brain.
On another note, my childhood friend use to run a trapline. He used to use an axe handle to finish off the muskrats and coons. One day he caught a possum and after about the 20th time of cranking it in the head and the darn thing just looking at him and hissing, he swore off the axe handle and started carrying his 22. Never had a problem after that.
BTW. Have you ever notice that possums have finger prints?