Eastern Europe

I spent 11 days in Budapest-Vienna-Prague, taking the train from one to the next. Krakow is quite a bit farther. Flying uses up a lot of time - airports are
far from city centers while train stations are in or near them. And you have to be at the airport 2-3 hours before your flight, then have to wait for your luggage at the other end, if you check it - and flights within Europe usually have very limited carry-on allowances. You might be able to take an overnight train - esp. from Prague to Krakow, but you need to be a really sound sleeper to benefit from that - otherwise you lose a night's sleep and end up cranky and unhappy the next day.

I think train is definitely best, but you are looking at some overnight trains. Or, stop in Vienna in between Prague and Budapest for a couple of nights. Vienna is a worthwhile stop for the palaces and art, at a minimum. Overnight trains are not bad at all. At least not the ones between Madrid and Paris, Paris and Florence (don't know if they still have that one), and Barcelona and Granada. Those are the ones that I have been on. I find the worst thing about it is that you wake up in your clothes and feel pretty grungy. It also seems that fewer night trains these days have two-person compartments. One thing to consider, a lot of the smaller cities are very, very worthwhile to visit. Namely Sopron, Hungrary; Olomouc, Telc, and Cesky Krumlov, Czech Republic. We were is all of those places earlier this year along with the big cities in those countries. I struggled a lot giving up Krakow and Berlin, but we really wanted to see the smaller towns, too. We were not disappointed.

You could also look at the buses - Orangebus, Eurobus, Student Agency.
Although we had a particularly bad experience on Orangebus between Budapest and Krakow this summer, that may have been an anomaly for that particular trip only, as we have seen reasonable reviews for them. The buses are way less expensive than the trains, reserved seat tickets can be ordered online in advance. The buses are faster than the trains as they can take direct routes that the trains cannot, e.g. Budapest-Krakow is 7.5 hours.

Krakow really doesn't fit geographically. Don't get me wrong, I really liked Krakow (much more than Budapest and Prague), but with your tight schedule I'd consider alternatives to Krakow or look at flights instead of trains/buses. You could do Budapest - (train) - Prague -(overnight train) Krakow and then fly to Rome, but other than that I really can't see you actually sightseeing in these 3 cities, travelling between them and getting to Rome with just 11 days to spare.

There are overnight trains connecting all three of these cities, so doing that, 11 days in those three is quite doable.