If you really, really want to see the castles of Neuschwanstein (the one on all of the brochures and postcards) and nearby Hohenschwangau, to answer your question - yes, you can get there by train so it's not necessary to rent a car and drive down there. You would take the train from the main Munich station to Füssen; it's about a 2 hour trip. From there the castles are only about 2-3 miles and there is a bus connection or you could rent a bike.

But it would consume most of your day and and you'd spend more than 4 hours on a train. I'd agree with the others and say that you should sample more of what Munich has to offer (especially the beer) and save the alpine castles for another time.