Usually he comes up to Stockholm for the weekends, then he stays in his appartment at Slottsbacken 2, opposite the Royal Palace. The Prince likes to go out nightclubbing and you can spot him at the clubs around Stureplan.
He goes to sport events, restaurants, both for brunch and dinner and for a coffe in coffeshops. Also he takes his dog for walks at Djurgården.
But, the clubs/restaurants around Stureplan is the best chance to see him!