Evangeline,
The other posters are right, you simply do not have time to visit both Rothenburg and Neuschwanstein. With only one full day in Munich, I'd stay in the city. There really is a lot to explore, including Nymphenburg Castle (more of a palace, really), if you are looking specifically to fit in a castle. You could easily spend two to three days exploring and that doesn't include visiting any museums.
If you are looking to visit Neuschwanstein Castle, you can get there very easily by train but my personal preference would be to rent a car (it really is pretty straightforward to get to and from the main train station, especially if you have a GPS). Or, if you really want to, hiring a private driver/tour guide. If you had a car/driver, I would suggest starting early and heading straight to Neuschwanstein. I'd visit Hohenschwangau, it really is the more interesting tour in my opinion, and/or Neuschwanstein. Then I'd head to Linderhof, King Ludwig's 'hunting lodge'. Although it may not resemble a fairy tale castle, it really is the best of the castles he built and he actually lived here for a while. If you had time, I'd fit in a visit to Wieskirche, Oberammergau, Ettal Monastery and/or Andechs on your way back to Munich.
I'd skip Rothenburg, it will take you about 3 hours to get there either by train or by car, then 3 hours back. There are other posters here who can recommend good alternatives that would be closer. If you do decide to go, I'd head straight for the town centre and then spend all your time exploring.
I hope this helps a little. Have a great trip!