I've been to Neuschwanstein 3 times since 1988, and I keep going back, but I never took one of those overpriced, third party tours.
The first time (1988) I was coming from four days in Munich to Immenstadt i.A. for the night before heading up to Oberstdorf and Kleinwalsertal. We made a side trip to Füssen. Since we were using a rail pass that day the side trip cost very little extra. Our only expense was the bus from Füssen and the castle admission.
The second time (2002) we were staying near Oberstdorf and came over on a bus. An Oberallgäu pass we got from our Gasthaus covered most of the trip at no additional cost.
The last time (2017) we were already staying for three nights in Pfronten, near Füssen, and used a bus pass from our Gasthaus.
Mike's Bikes charge almost a hundred dollars and doesn't even include the cost of castle admission. Other third party tours are similarly priced. All they really do is babysit you on the trip out to Füssen; you can easily do it on your own. You pay extra for the castle tour, which is given by someone on the castle staff. Having gone to the castle three times myself, without a tour, I would never spend the extra money for what I consider a needless service.
There are a lot of things to do and see in Munich. If this is your first time to Munich, I think it would be a shame to skip all those things for a long day trip. Stay in town.
I'm an engineer and kind of a technology geek. I really enjoyed the Deutsches Museum. I also enjoyed the Münchner Stadtmuseum, but there's no English. All the exhibits are explained in German (of which I understand a little).