OK I took the train and bus going to and coming back from Schwangau to see the Neuschwanstein castle but I joined the tour group. It's much convenient though a bit over than going on my own.
Radius Tours - where I booked my castle tour is one of those popular tour agency in Munich, heard about them previously but I totally forgot, then again, the couples I went with in Rotehnburg are going with them and I remember and I went to their office right inside the Munich main train station between platform 27 - 36, on the morning right before the tour. No advance booking necessary.
Our tour guide - Mike - is a young comic guy, making jokes every now and then and giving info during the pleasant trip. He made sure that no one is missing in the group and that we all - about 30 or sumthin of us in the group - board the same train or bus on the same carriage.
The Neuschwanstein Castle tour starts around 10 at the munich train station and end late in the afternoon, costs EUR35 + 9 castle entry.
They also have other tours in the area and a bike rental also.
If you want to see both castles (Neuschwanstein, Hohenschwangau) is it maybe better and nicer to drive with a horse-drawn carriage to the castles. It is easier because the castles in the mountains.
A little bus drives also to Neuschwanstein.
You find the horse-drawn carriage and the bus not far from the ticket office.