Amtrak continued to run during the last shut down right? Why should this be any different. To bad airlines won't shut down though
Amtrak operates on either tracks they own, owned by the state / local transit agency for the NEC and some other corridors or tracks owned by private freight companies. Same for all for the dispatch and control operations. Amtrak provides it's own security operation with the Amtrak police, although at a few major stations such as NYP, there is additional security presence provided by local police or TSA. No reason for Amtrak to stop running for a short term federal shutdown.
Airline operations, however, are dependent on the federally operated Air Traffic Control, TSA and probably other parts of the Home Security Department and Customs for international flights. These presumably all would be covered as Essential Personnel, even though all essential personnel are also unfunded if Congress doesn't pass an authorization bill or continuing resolution.
I think odds are good that we will have a short term government shutdown. The Republican House leadership has not shown that it can effectively control the newly elected block of 50 or 60 idealogical Tea partiers who are new to the political process. The House Democrats are not going to be inclined to help the Republican leadership out, if that means draconian budgets cuts out of a relatively small slice of the federal budget. Short term gridlock is likely.