No, you cannot do the railway routing with the StreetMap since it is mostly suited for street network transportation purposes. Even though StreetMap contains railway tracks features, they are not modelled appropriately to support this type of application.
Yoy might like checking some user railways showcases on Esri's web page. You perhaps have heard about Network Analyst extension for ArcGIS where you can create multimodal networks (to include car/public transport roads, pedestrian ways, cycling, subway lines etc.). Network Analyst is definitely capable of doing railway routing and I believe there should be some successful stories on that available from Esri - you might get in touch with a key account manager working with railway industries.
Another thing you might check is Geometric Networks which are available as a core functionality of ArcGIS product. It is quite different from Network Analyst yet it has certain benefits railway administration users might find very useful. In any case, you will likely need either to collect the railway tracks data as a vector dataset and perform some geodatabase modelling.
