places to visit before or after a danube river cruise

Have you considered continuing south on the Danube from Budapest, ending in the Black Sea?

I had not. I will check into that. Thanks for the tip.

Several river cruise companies in addition to Viking offer the lower Danube adventure. Each have their merits. Having traveled the length of the Danube, it is my opinion that the lower half is far more culturally interesting.

I've been on a Viking cruise (Rhine). I made my own flight arrangements (and actually the air was just a bit more than Viking's airfare for a better itinerary) Because I made my own arrangements, I did my own "add ons" for much less money and I could do anything I desired so I added extra days at both ends of the trip. For you, Budapest is a wonderful city and you could stay there a few more days. I'm not sure which Danube cruise you are taking but neither spends that much time in Budapest. From Budapest, you could visit Bratislava for a couple days and then go on to Prague. There is plenty to do in Prague, and also many smaller cities in the Czech Republic you could visit.

Prague and Berlin are good choices to visit before the beginning of the cruise, as they are relatively well connected to Nuremberg by train/bus. (I am wondering, though, how you can start a cruise on the Danube in Nuremberg - the town is not on the Danube?). The same is true for Munich, Bamberg, Regensburg, and Erfurt. These are the ones that immediately came to my mind - though, with two weeks, you could probably go anywhere in central Europe...
Any special interests?

Nürnberg (Nuremberg) is on the Rhein-Main-Donau Kanal (Rhine-Main-Danube Canal), that connects the rivers Main and Danube.
Before the breakdown of the Eastern Bloc it was named by someone the silliest structure after the Tower of Babel.

Nürnberg (Nuremberg) is on the Rhein-Main-Donau Kanal (Rhine-Main-Danube Canal), that connects the rivers Main and Danube.

Thanks for pointing that out. Had not thought about that - I feel silly now.

Nuremberg is a great city to spend some extra time, and you can also do an easy day trip to Bamberg. Definitely consider continuing to Berlin and/or even Dresden. (My wife and I did the same route on a river cruise in the opposite direction a few years back. We stayed over in Nuremberg for a couple of nights, then Munich for a couple of nights. Spent two nights in Budapest on the front end. It was our second visit to all three of those). Munich would also be a practical choice, but really Nuremberg is pretty central that with two weeks the possibilities are endless. For example, you could start in the Rhine/Mosel region. Prague and Krakow would be good choices too.

Thank you all for those great ideas. Very much appreciated.