From Berlin, the trains arrive around lunchtime of day 21, give or take an hour. After getting to the hotel and changing money (what, they use Korunys instead of Euros? Of the places I visit, the UK, Denmark, Sweden, Czech Republic, and Switzerland still use their own money, while France, Germany, Austria, Italy, Spain, and Portugal are on the Euro).
Here's a suggested itinerary:
- Day 1: Walk the Royal Route (or at least part of it). From the top of the castle hill in Hradcany, tour Prague Castle in the morning. After lunch begin your slow descent through the odd hill-bound architecture of Lesser Town (Malá Strana), including the St. Vitus Cathedral, The Royal Palace, St. George's Basilica, and the Golden Lane. Then stroll across Charles Bridge, on the way to the winding alleys of Old Town (Staré Mesto). You can happily get lost finding Old Town Square (Staromestské nám.), stopping at private galleries and cafes along the way, including the Old Town Hall & Astronomical Clock. From Old Town Square take Celetná street to Ovocný trh, and you'll reach Mozart's Prague venue, the Estates' Theater.
- Day 2: On your second day, explore the varied sights of New Town, Old Town, the Jewish Quarter, and Lesser Town -- what you didn't have time for the day before. Just wander and browse. Sights they suggest include Wenceslas Square, the National Museum, Alfons Mucha Museum, Municipal House, Powder Gate, the Jewish Museum, the Charles Bridge (again) at sunset, and the Kampa. Not sure about all the museums, but I really like the idea of a day of just wandering around.
- Day 3: either stay in town and do stuff like a 3-hour cruise on the Vltava at sunset, or take a day trip to Karlstejn Castle. The Castle is a 45-minute train ride from Prague and is open 9am to noon and 12:30 to 5pm so I can probably do that during the day and then maybe be back in time for the cruise in the evening.
There's an overnight train from Prague to Vienna that leaves just after midnight and gets in after 6am.