I think I followed the crowd to Bad Canstatt station. Looking at Google maps, 'Neckar Park' station is closer and also brings you out at the away end section, but I think the line splits after Bad Canstatt so the trains aren't as frequent to Neckar Park.
For some more general advice, I use this website a lot for my Bundesliga travels. If you play around with it a bit, there's another page with a bit of advice about the city of Stuttgart too
http://eurofootballcities.com/index.php/cities/stuttgart/vfb-stuttgart
http://eurofootballcities.com/index.php/cities/stuttgart/vfb-stuttgart
Sounds like a bit of a ball-ache to access the away section, but at least it sounds like the beer is alcoholic.
The away section is the pink corner below. Give City a few days to sort themselves out and I'm sure we'll be able to get tickets through the City website.