You can download this package of tools from ArcGIS Online to create a network dataset that uses GTFS transit schedule data.
http://www.arcgis.com/home/item.html?id=0fa52a75d9ba4abcad6b88bb6285fae1
The download includes instructions on how to set it up.
However, you will have to have an editable street feature class that you can use as the basis for your network dataset. You can obtain streets from your city, county, or metropolitan planning organization, OpenStreetMap, or from Esri's Streetmap Premium product (a newer file geodatabase version rather than the old SDC version you're using).