I think there's a second operative question here, too: If a new Congress is elected that is heavy to either party and the "new guys" can get a deal together but can't implement it for a few weeks after the deadline and the old Congress won't budge, what happens during "limbo"?
How the looming multiple deadlines of December 31 will play out is anyone's guess. The budget sequestrations are not that big a deal in the immediate term because the $1.2 trillion is spread over 10 years. What the budget cuts do is create uncertainty for the federal agencies in planning and funding for grants, contract awards, etc. But federal agencies have gotten a lot of experience in dealing with short term budget uncertainties in recent years. What sequestration would mean for Amtrak, really don't know, but Amtrak as a private company with its own revenue stream has a cash reserve which would allow it to keep everything running for a while if there is no deal in Congress.
The big problem is the expiration of the Bush tax cuts. Clearly they will not all permanently expire - although that would make a substantial reduction in the budget deficit. But the whole mess creates a lot of uncertainty in the private sector.
Back to your question, the new House and the newly or re-elected Senators would get sworn in on January 3. If the House returns to Democratic control with a narrow majority and Dems maintain control of the Senate & White House, there may be no ground to reach an agreement in the lame duck session, even on a temporary extension, so everything expires on December 31. Then the Democrats would try to hammer out a deal with the Senate Republicans who will still be in position to block everything and pass a bill, maybe late January, retroactively setting the tax rates and budget back to January 1.
If the Republicans retain a narrow control of the House and Obama is re-elected, then there may be a deal in December after much hue and cry.
If Romney & Ryan are elected, the Republicans gain control of the Senate, retain control of the House, then Amtrak's situation is much more perilous and complicated than a mere budget sequestration. In that event, Amtrak's LD trains would be at risk of getting shut down and probably lose some state supported routes as well.