"Did Neanderthal man possess a soul"
Well first you have to make the assumption that the 'essence' of yourself is a soul, and than ask if other species are the same in this regard. Since I have to take on trust other human beings are like myself, and not simply automatons that appear to be like me, then presumably I can make the same assumption of other species or sub-species.
"if he did what do Christians believe happened to him when he died"
Well having a soul doesn't automatically mean a Christian viewpoint is correct, but assuming they are in the right ballpark, I guess it would depend on which particular Christian you asked. I'm unsure if there is an official church viewpoint. But I'd suggest probably the same experience would occur to them as occurs with humans when they die. I suggest that because the Neanderthal would have a reasonable amount of intellect allowing them to make moral decisions and question their own existence so can't be too far removed from humans.
"I was taught that no one can go to heaven unless they had been baptised"
I'm surprised you were not also informed about what happens to humans that have never heard of the Christian religion. They are "in a similar boat" to your hypothetical Neanderthal. I believe the explanation is that they are not held responsible for things they have no knowledge of so get a "Get Into Heaven" card as default.
Of course, alternatively, you could always consider that maybe there is a flaw in the assumption that only baptised folk go to heaven. Do you really think a deity is going to care whether their creations have gone through some made up ritual or not when making a decision as to what happens to them next ? Is it not more likely that this claim is a control tactic by the church to get you to do as you are told, preferably unquestionably ?