Sounds like a great itinerary. Neuschwantstein will be a whole day excursion. I have to admit that I find the castle far more impressive from the outside than from the inside where it is not finished, not really old and it is quite a mass event (guided tours of 50 people, one followed by another every 5 minutes or so). But the setting is beautiful and if you have time to also visit the Hohenschwangau Castle that is close to the ticket office, I can recommend that. This used to be the summer castle of the Bavarian kings and feels much more lived in than Neuschwanstein.
As to Munich, since you don't have a lot of time, I would just recommend a walking tour through the city center: Marienplatz, Viktualienmarkt, have a look at the Hofbräuhaus. One store that is great fun is Manufactum:
https://www.manufactum.de/manufactum-muenchen-c-43/ Also fun to walk through is the shopping arcade Fünf Höfe
https://www.fuenfhoefe.de/?L=1
As for restaurants, I like the Schneider Bräuhaus
https://www.schneider-brauhaus.de/ and the Augustiner
http://www.augustiner-restaurant.com/html/english.html Both are in the city center and are very typical Munich guest houses. While they do cater to tourists, they also are very authentic and locals would go there, too.
A little less in the immediate city center is the Wirtshaus in der Au
https://wirtshausinderau.de/en/ which friends of mine love and I have sent quite a few people there who all loved it. Haven't made it there myself though. But so far everyone really liked it.
If the weather is nice and you want a beer garden, there are numerous ones, but the two that are really popular with locals are the one at the Chinesischer Turm (Chinese Tower) in the Englischer Garten (English Garden, the big park in Munich):
https://chinaturm.de/biergarten/index.html and the Hirschgarten (deer garden, this is where the king had deer held for hunting, now it is a park):
https://www.hirschgarten.com/beer-g...bavaria-koeniglicher-hirschgarten-munich.html
A Munich beer garden is special as it is not just any outdoor seating area, but it was started by the breweries. They planted chestnut trees above there beer cellars where they stored the beer, because the chestnut trees grow very large and give lots of shade and that kept the cellars cooler during the summer. People liked sitting under those chestnut trees and therefore the breweries thought that this was a great idea to make additional money and they started to sell the beer directly to the customers. This upset the local pub and guest house owners as it created new competitors and therefore they lobbied that the breweries should not be allowed to do that. The compromise that was drawn up was that the breweries were only allowed to sell beer, but no food. So people started to bring picnics with them while drinking the beer. Now beer gardens are allowed to sell food, but a true beer garden will still allow guests to bring in their own food and you will see whole families bringing table clothes, dishes and their own food while purchasing the beer. Classical beer garden dishes are giant pretzels, Obazda (a mixiture of camembert, butter and paprika with onions) and garden radish (not the small red one, but rather large and all white, but same taste).
Have a great trip!