Take the cogwheel train from Alpnachstad (accessible via Luzern) to Mount Pilatus, enjoy breathtaking views and hiking trails; the train runs from 8:30am to 5pm.
Catch the train and bus (~1.5h) to Vaduz, Liechtenstein to explore the Prince's Castle (outside view), Vaduz Art Museum, and local shops; museums typically open 10am-5pm.