A teleporter may not solve your financial problems because they cost quite a lot to build.
There is a simpler train route via Vienna and Bratislava which has the advantage of overnight journeys from Rome to Vienna and from Bratislava to Krakow, with a full day site-seeing in either Vienna or Bratislava. A travel pass would make it a comparable cost to flying.
I found
http://www.raileurope.com/ much more helpful than
http://www.raileurope.co.uk/. The UK site just said it couldn't give me the details that I wanted.
One other thing about Rail Europe. They have a limited supply of tickets and so they sometimes tell you that the train is fully booked even when it is nearly empty.
ETA:
Trenitalia gives you much more information about trains from Rome to Vienna, and faster ones than I found on Rail Europe. It's beginning to look as though the train is feasible.
And for a Slovakia train schedule, try
http://kvc2.zsr.sk/EN/.