Mount Tegelberg is located several kilometers from the Neuschwanstein Castle. So, if you come to the castle, why not observe a great view of the surroundings from the top.
From the castle find a staircase and then the path (here it is). Get down along it and look for signs with a cable car (Tegelbergbahn). The walk takes about 30 minutes through the beautiful alpine fields.
While lifting you can watch excellent view of the Neuschwanstein Castle. There's a lot of space at the top, you can easily walk in any direction.
Near the lower cable car station is a Rodelbahn (something like summer bobsleigh).
You can climb the mountain on foot, here is the route map:
Map of a walking route to Mount Tegelberg
Webcams on Mount Tegelberg
Getting there: You can get to the cable car on Tegelberg (Tegelbergbahn, Schwangau) from the train station in Füssen (Füssen Bahnhof) and the castle (Hohenschwangau, Schwangau (Schlösser)) by busses #73 and #78. You can also walk from the Neuschwanstein Castle. Opening hours and prices are here.