Before and after Berlin-Prague-Vienna in Sept 2018

I don't see this on your tour in Berlin, a guided tour of the Reichstag building, you have to fill out the registration form and submit it by email no more than 3 months in advance of your desired date.
https://www.bundestag.de/en/visittheBundestag/dome/tours

Many hotels in Europe just don't book that far out. To test that, try to book for a date a week before and a week after Oktoberfest - if they have availability, then I guess they are booked. But if not, they probably just aren't taking reservations yet - may not even have 2018 prices set yet.

I was not especially impressed by Munich, but I found a lot to like in Frankfurt. Consider skipping M and spending a couple nights in F - lovely old town center, and daytrips to nearby towns that are interesting and charming.

I took this tour in 2015 and thoroughly enjoyed it. We arrived a day early and went to the Pergamon Museum and the art museum on Museum Island- I can't remember the name of the art museum - there was a special exhibit that we saw and loved of a comparison between the Impressionists of Paris and Expressionists of Germany.

The Reichstag tour was interesting to some of our tour mates who scheduled it for free time on one afternoon. We went to the Jewish Museum on our free time which was very good. We also saw the wonderful Gemaldegalerie which had a Vermeer as I recall. There are many options in Berlin depending on your interests.
Vienna: our tour did not go to Schonnbrunn Palace so if that's of interest to you, you could take a day post-tour to do that. Or, go to Salzburg before you head to Germany. All good ideas!

You have tons of time to prepare. Check a guide book to Germany out of your local library and read about what there is to do in your starting and ending cities. Berlin, in particular, has a large number of fabulous museums plus fascinating historical sights. I loved the Cold War stuff; you might not. You should choose things that sound interesting to you. There will be no shortage.

Unfortunately, your "one day early" in Berlin will probably be significantly affected by jetlag and sleep-deprivation. You may find that you're too mentally foggy to take in a museum, so I'd be extremely hesitant about trying to buy an advance ticket to a costly museum. And the lines for the Museuminseln museums can be over an hour long. I suggest deciding on one or two things you believe might be especially interesting but having a back-up plan if you're only up to walking around. There are walking-tour opportunities, for example. There are also boat trips on the river, but I would fall asleep on my arrival day if I sat down on a boat. Berlin is not a particularly gorgeous city, so I think a walking tour might be better than just randomly striking out on your own.

As far as the route from Vienna to Bamberg, Passau is about half way, and Regensburg is half way from Passau to Bamberg.

I'm sorry I can't help with Berlin. But, I agree with Acraven that going in one day early, the arrival day probably wouldn't be a good time for a museum. Hubby and I always go in 2 days early since our first trip to Europe. Arrival day free time would be a good time to walk around the hotel neighborhood, get a coffee or beer, do some outdoor looking around, stay out in the daylight. I like the idea of a walking tour. First trip to Europe we got on a HOHO bus the arrival day and I slept on my hubby's shoulder the whole time!

If you go to Munich there is lots to see. The one thing that we found, a friend directed us to, was a very small museum/display at the University. The White Rose museum about Sophie Scholl, her brother and friends. University students who worked/resisted against the Nazis and were executed. It was very moving. There is no admission fee, is very small and has a jar for donations. We all thought it was interesting.

Thank you all for your suggestions! We are actually arriving 2 days before the tour starts in Berlin. The jet lag day will be spent on the train from Frankfurt to Berlin, so I didn't count it. (Non-stop flight from Austin to Frankfurt). It will be evening when we get to Berlin - time for a nice dinner and relaxing into the hotel.

Post-tour, Munich is looking more promising. I checked Homeaway and found several reasonably priced and well located options. Will get one of those booked quickly.

Everyone knows where I will be going with this.

The tour ends on Thursday. We'll be working our way back to Frankfurt
for our return flight on the following Monday.

Spend all day Thursday seeing what ever you think you missed with the tour in Vienna. But stay flexible. You might discover that Vienna isn’t what you thought it would be. Not picking on Vienna, I try and keep all my trips a bit flexible as time is precious and I never know what to expect. Anyway, Friday get up early and hop a train to Budapest. 2.5 hours and about $30. You are just too close, its too cheap and too easy to not do it.

Depending on your exact dates you might be there for the Jewish Cultural Festival or the Buda Castle Wine Festival or there is a pretty odd, but fun, pig festival in September. And the weather is stunning in Budapest (and Vienna) in September / October.

That will give you most of Friday, and Saturday and Sunday in Budapest. Monday morning Lufthansa has early morning flights to Frankfurt for about $130 one way. I’ve been on one or another of the flights for Frankfurt connections to the US on a number of occasions so if your connection is after 10am this will work. If your flight out of Frankfurt is earlier, Wizzair has an afternoon flight the day prior.

Apartment will cost you no more than $60 in an excellent location..... Figure overall 10 to 15% less than Austria or Germany.

Take a boat ride on the Spree River on your arrival day. You will get views of many important places and won't need to concentrate on museum collections.

"Everyone knows where I will be going with this."
Yes, James, I knew.

I resisted for an entire day.

We took this tour last year and loved it. We arrived 3 days early in Berlin and still didn't see everything on our list. We especially enjoyed our time in the Reichstag dome. We arrived just before sunset as a thunderstorm was clearing. It was magical. Make sure to book well in advance. I booked about 2-3 weeks ahead and the only time available during our 5 days was the evening of the first day of our tour.

It looks as though the tour now includes the Berlin Wall Memorial. We did that on our own. Well worth it.

We only had time for a few museums, but the Pergamon was our favorite.

We spent our extra day in Vienna at the Schonbrunn Palace. Easy subway trip out there, and lovely gardens. Go all the way up to the Gloriette. Great view back down.

We enjoyed our wander through the Stadtpark and St. Stephens Cathedral with a ride up the elevator of the south tower.

I took this travel in 2011. My pre-tour days I did a late afternoon pre-arranged Reichstag tour; spent lots of time at the Cold War exhibits along the EastSide Gallery beginning in the morning; visited the Topography museum. One museum you will not see in one visit is the German History Musuem, excellent. In 2011 I opted to spend free time at the Pergamon Musuem and the Jewish Museum which I had not seen previously. *If the tour is the same you have two afternoons for free time, one day is longer than the other.

A favorite activity is just walking in different neighborhoods without being guided. There is so much to be immersed in. Berlin is vibrant, friendly, and hip. In 2006 we stayed in the western side area of Kreuzberg, and in 2011 the tour stayed in the former eastern neighborhood of Prezlauer Berg. I enjoyed the former eastern area more.

Berlin is one of those cities where everywhere you look you see something of interest so it's good to have an idea of what your top priorities will be for your free time both before the tour starts, and your free time during the tour. Depending on your budget utilize your 3Day musuem pass you'll get from the tour as much as possible. I bought an additional welcome pass card which was good for the trams and buses, and admission into the museums. It was convenient and allowed me flexibility.

Look for the stepping stones whenever you are in Europe. They memorialize Jewish citizens who were deported. Enjoy!