You must try:

• a Biergarten
• a swim in the river Isar
• the public parks in the city (new york central park is just a joke)
• climb up the stairs of a church tower and enjoy the view, preferably the 'Alter Peter’
• public transport in Munich, especially the trams
• window shopping in the Maximilianstrasse

If you go to Neuschwanstein, take your time and enjoy a whole day there, first do some hiking in the area around, 2-3 hours, then walk up to the castle itself, and afterwards a little boat trip on the lake Forggensee nearby.