Favorite thing: The main city library is designed by Danish architect Henning Larsen and is one of the most spectacular modern buildings in the city. It is also a great place for checking your mail as it is free - all you need to do is book a slot with the librarians and then you have limited time but enough to keep in contact with people.
Checking your email and surfing the net can be done several places, but it's hard to beat the free service at the library. That is if you can do with 15 minutes.
Check out the opening hours here. The page is in swedish, but you should be able to distinguish the times at the top just to the right of the middle.
Favorite thing: Everybody (well, almost!) likes receiving letters - I certainly do, though there are other, more modern and efficient means of communication - e-mail, phone, and such like. However, if you simply like getting letters or if you need, say, the originals or paper copies of some documents sent to you while you are on the move, you can use the poste restante service. Your best bet is to use the American Express service, which is available for free to clients - you should hold either their card or travellers cheques. They are reliable and present in all major cities.