If you don't get a chance to leave Sapporo (or the station for that matter), the 10th floor of the ESTA building, within the confines of Sapporo Station, has a ramen park called Sapporo Ramen Kyowakoku. It is here that you'll find 8 ramen shops from around Hokkaido ready to serve the best ramen in perhaps all of Japan. Choose carefully and vote wisely.
The 8 shops will change every year or so depending upon your vote. The current lineup as of April 2009 can be found here. There's also a gift shop where you can find all of your ramen goodies. It's the next best thing to traveling throughout Hokkaido for ramen.