Neuschwanstein

Robert - While it has been some years since I visited (2012), I think I spent more time waiting and hiking up than touring, especially Neuschwanstein.

If you do both they give you a timed ticket to one of them (for me it was Hohenschwangau first) then a timed ticket to Neuschwanstein for a few hours later (maybe 2 and half).

The tours are led by a guide and it seems they have to keep to timed schedule because the group behind is only a few minutes behind. I think in total we walked around maybe an hour or so in each castle.

Be forewarned that it does take some time to get up the hills to each of the castles even for somewhat fit people. For Neuschwanstein it says on the website 30-40 minutes but I think it took over an hour for me because I had to stop several times to get my breath back and while I am not the most fit person I am not out of shape either.

There is a "bus service" to the back of Neuschwanstein but its not on a schedule.

I did not tour the Museum.

Edit: I looked at the website and it says the tour of Neuschwanstein is 30 minutes Hohenschwangau is 35 minutes.
https://www.hohenschwangau.de/1401.0.html

Another Edit: So to answer your question with another question about the afternoon or next morning do you think you'll leave Rothenburg early enough to get a timed ticket for the early afternoon, since they close at 6 so the last tour you could get at the first castle and there would be enough time to get to the second and tour the second before it closes?

One plus is there will be very little truck traffic on Sunday. Trucks are prohibited from driving on the Autobahn on Sunday unless they have a special permit. Baring a Stau, the dive to Füssen should not be too stressful. Ski season is over and school vacation time is yet to begin (Pfingstferien the first week of June) . With an early start you should be able to see at least one of the castles on Sunday.

I would go on Monday. There are plenty of places you might want to stop and visit on the way to Fussen.

I think I would prefer to stop on the way.

Hi Robert: It is now early may and I wonder if you have done this journey ... we are contemplating somewhat the same scenario in the middle of June, would arrive in Fussen on a Sunday afternoon for an overnight stay. We would be in Fussen Monday morning and then will be heading to Munich after lunch. Let me know how this all came together. Thank you, Rita

Hi Rita,
Sorry for the delay, but we were still on our trip when you posted your question. Here's what I can share with you about my recent trip to Fussen and Neuschwanstein.

The drive from Rothenburg ob der Tauber on a Sunday was fairly painless, as pointed out by MCHPP, the lack of traffic (trucks) made for an easy drive with no hassle at all. We gave ourselves plenty of time and arrived with time to spare. Also, RS's advice to skip the early parking lots and to drive past the ticket center to parking area 4 was a well worth it. We parked with ease and were in the queue to pickup our tickets in moments.

I will say this.. Neuschwanstein was major letdown compared to everything else we saw and did during our trip. The Tour there is very sterile, and you are rushed through from start to finish in 25 minutes. It's fitting that Walt Disney used this as his inspiration for his theme park, because that's where we felt like we were. Towards the end our our quick run through the tour, I had the man in front of me lean over to his wife and mention that he thought it was the worst tour they had taken in Germany as well. It turns out they are avid RS readers and followers as well. We were so let down that we skipped the hike up to Mary's Bridge in favor of getting out of there instead.

I do not say this to bring a cloud to your plans, but to point out that if Neuschwanstein isn't something that you are absolutely locked into doing, you may wish to save yourself the trouble of going all the way to Fussen for this one thing, as we found several other castles and historical sites far more interesting in other parts of the country.

Regardless of what you decide to do, I sincerely hope that you have a wonderful trip. My wife and I absolutely loved our Germany adventure and already can't wait to go back.

Best,
Robert