Being in Prague feels like a time machine just sent you back to around year 1800. You have to walk pretty far out of the old city and Mala Strana to see the newer world. The exception is off course the many tourists, content of the shops and so on. You also have many areas without cars, it makes this feeling even stronger. Prague is probably the city I've been in that is most true to the old architecture.
Having one day in Prague (combined it with a business meeting) I chose to mainly walk around, just taking in the medieval surroundings, taking picture and eating amazing trdelink! The Old city and Mala Strana are the best areas, and Charles bridge! (just make sure to pass it early in the morning, too much people the rest of the day). You can't take the wrong direction in these areas, old beauty combined with candy colored buildings everywhere.
I also went to the part of the city called Vrsovice, named the hipster part of the city. It's not quite Williamsburg, but definitely has it's very cool cafes and all of that. Krafarna is one good choiche.
Do not bother going to the Black light theatre they talk so much about. Not very impressive, mostly strange, I left before the second half started. But you should eat at St Martin, KARE cafe or U Mecanase. My hotel Vintage Sax was very central and pretty cool as well.
Enjoy medieval times and don't forget to eat that trdelink!