If I had time enough in Stockholm to take a day trip, where should I go? Sigtuna? Birko? The Archipelago somewhere? Are there prominent Iron Age, Stone Age or Viking Age sites close to Stockholm?
Sigtuna is a small charming town that makes a nice day trip. But if you are looking for Viking age sites, the obvious answer is Uppsala, 30-40 minutes north of Stockholm. Gamla Uppsala (Old Uppsala), was the big political and cultural centre in the area during the Viking age. There are three large burial mounds there, as well as a small but great museum about. There are also a huge iron age findings just outside Gamla Uppsala, the items can now be seen in Uppsala's university museum.
In addition, there are a huge number of rune stones in and around Uppsala. If Bronze age sites are of interest there is also Bronze age burial mound in the western parts i Uppsala. Other prominent burial mounds are Ottarshögen a bit north of Uppsala and Anundshög outside Västerås, both late iron age.
There are several sites I think are Stone Age, but I don't think there are any that can be called prominent.
The archipelago is nice, there are many great options for both longer and shorter trips.
Uppsala sounds very attractive. I have always wanted to see a runestone in its original setting, rather than in captivity in a museum.
It is certainly a great place to see free range rune stones! Although the stones are on average around 1000 years old so many have been moved for different reasons and it can be hard to know their original location. But seeing them in the landscape is very different to seeing them in a museum.
If original location is important, it sometimes happened that the runes where carved into the bedrock and not on loose stones. In that case you can be sure that the carving is in the original location. There are two such carvings in southwestern Uppsala that are pretty easy to access.
And if you are looking for charming small towns I should probably add Norrtälje and Öregrund to the list.