I spent a day there after a business meeting so did not have a huge amount of time but there are several narrow streets leading off the Old Town Square, or up towards the castle. Those, along with the Old Square and Charles Bridge, kept me occupied for quite a while.
Three things I would recommend are to take a tripod for photos of Charles Bridge at night (or early morning) - an absolutely fantastic way to while away the time; go to the Old Town Square for the evening market; and go up the bell tower in the Old Town square for a view over the city