I'm not quite sure. Neville's dealt with a lot during his life and while he did try and hide everything when he was younger, everything seemed to make him all the more determined to do better later on. He was cheerful and still worked so hard despite his difficult early life. In many ways, I find Neville to be stronger than Harry.
Given the right circumstances, I don't think that it's impossible that Neville would develop a drinking problem, but I think that it would take more than it would to make Harry one.