A few thoughts from my experiences:

1. Different Amtrak trains are slightly different.
a. Some you have spots in the cars themselves to put bikes.
b. Some have a roll-on/roll-off service where bikes go in the baggage car but you bring them there without a box as long as you have reserved a spot.
c. Some may still have bikes in baggage car in box.

Most of the trains I've ridden, are in category c but have moved to category b. In both cases, you only get access to the baggage car at designated baggage stops. At those stations, you also have an attendant available before the train arrives who can sell tickets - so there isn't as much a notion of buying a ticket on the train.

I've used Amtrak to keep my options flexible but haven't specifically waited to show up at the station to buy a ticket. Instead, I've:
- Bought a ticket a few days in advance via the internet, since at that point it was pretty predictable when I was leaving.
- Called Amtrak before to change dates of my reservations. While I haven't done that right before a trip, I have assumed I could try this if necessary and be subject to whatever was available. Similarly, I've assumed I might change the station by cycling to a further station. Many years ago when there was a train up the east coast, I did a variation of this. I wasn't sure if I was going to end up in Jacksonville, FL, Savannah, GA or Charleston SC. So I bought a ticket from Jacksonville back north. When I ended up in Savannah instead, I changed the ticket.