Are there certain days of the week that are best to visit Neuschwanstein for avoiding crowds in late May or early June?
Weekends are busiest. But if you show up early or reserve your tickets, crowds are mostly irrelevant. You may have to wait a little longer for the perfect photo op from the bridge or from the viewing platform below the castle, but that's about it.
Thanks to Tom for sharing his experience with us, he lived there for a few years.
There are a few big holidays in Germany around that time. For 2015 they are:
May 14 - Ascension day
May 24 - Whit Sunday (Pentacost)
May 25 - Whit Monday (2nd day of Pentacost)
June 4 - Corpus Christi
I don't know how busy these holidays are at Neuschwanstein, but I was in Berchtesgaden for Ascension day and the schools were out and the buses ran on a holiday schedule. Later I was in Dresden on Whit Monday. Whit is a big three day holiday in Germany, and Dresden was very crowded.
Not sure the ages of your traveling party and whether you like outdoor stuff, but we took the Bus Bavaria tour with Mike's Bikes and loved it so much.
We took the bus there, biked around, rode the Tegelberg alpine slide, had lunch, then to Mary's Bridge, and ended at the castle. A truly splendid day!!
Kim