Most trains between Munich and Salzburg take slightly longer than the 1h 24m you suggested. (The fast, but relatively infrequent, ones do take about one and a half hours but the majority take about an hour and three quarters. All services are direct, with no changes). However the train would definitely seem to be the easiest way to get there.
https://www.bahn.com/en/view/index.shtml
Getting to Neuschwanstein Castle on public transport is somewhat trickier. There's an hourly train service to Füssen, with alternate services being either direct or requiring a change of trains at Buchloe. That part of the journey takes around two hours. Then you need to take a 9651 bus from Füssen to the stop for the castle outside Schwangau. Although the bus ride only takes 8 minutes, those buses are very infrequent:
http://www.rvo-bus.de/oberbayernbus/view/mdb/kursbuch/mdb_266217_9651_101217.pdf
So a taxi might be better. (From there you have to take the path to walk to the castle, taking you about 20 minutes).
A few links for you:
https://www.muenchen.de/int/en/tourism.html
https://www.salzburg.info/en
http://www.neuschwanstein.de/englisch/tourist/
You might also wish to consider visiting Lindau, on Lake Constance:
https://www.lindau.de/en/
The express bus service is actually quicker than the trains, getting you there in a little over 2 hours:
https://www.flixbus.co.uk/bus-routes/munich-lindau