Wow, you have been experiencing lots of things that I have never experienced.
From Shin-Sapporo to Sapporo, the subway-Tozai line (East-West line) is available. But you need to change the subway at O-dori station to Nanboku line (South-North line). Also you can go to Susukino that is downtown of Sapporo city.
Sapporo is famous for fresh fish, sushi, and beer. You can enjoy draft beer and Genghis Khan food (BBQ lamb meet ?) at Sapporo beer garden and Kirin Beer garden.
http://www.kirinbeer-en.co.jp/menu1.html
http://www.sapporo-bier-garten.jp/
Sorry, only Japanese.
But I do not recommend to go there by yourself, so I do not tell you on PM and I know you do not drink a lot like me.