There are quite a few ways to get to the hut. The easiest would be to go from Oberjoch, resulting in a ~4 mile round trip hike with ~1,300 ft. of elevation gain. We recommend going up to the Hirschberg first, to take in the views of the valley.
The link below describes a fairly long tour, but the quickest way would be to take the bus from Hinterstein and get off at the trailhead for the Schwarzenberghütte. From there it is 3.2 miles with 2,800 ft. of elevation gain to the lake. If you're up for a little more climbing you can loop over the "Großer Daumen" summit. I would recommend going counter-clockwise from the lake.
If you plan on visiting Munich, it's worth stopping here. Perched above the Ammersee (great for swimming in warm water with alpine views!), the Benedictine priory and abbey is also home to a really great brewery. Walk up the cobblestones to the Baroque abbey and take in views of the Alps (if it's clear).
You can park for free along the street shown on map, and walk down to the shore. There are lots of little stony beaches and docks to set up camp. If you walk north there is a grassy meadow by the boat club. If you walk south there is a little restaurant with views of the castles in Füssen.