No one has answered so I'll give it a shot, but I should start by saying that I am a layman, although one with a interest in history. The following is more or less stuff you learn in school in Sweden. Years and numbers are from Wikipedia and are hopefully at least somewhat correct.

Early Swedish history is difficult because there are very few reliable written records, so historians have to rely heavily on archaeology. We know that people have certainly lived in what is now Sweden for a long time but to live in anything we could call a town is a relatively recent thing. During the Viking Age people generally lived in isolated farms or small villages of a few houses. Traditionally credited with being the first Swedish town is Birka, which was founded around the middle of the 8th century and had about 700 inhabitants at its peak. It was an important trading hub with links to Nogorod and the Byzantine Empire and it was visited by Christian missionaries, of which we know at least something through German writings about them.

There are other settlements which may be called towns that existed before or at the same time as Birka, like Uppåkra in southern Sweden (the part that was Danish until the 1600's), but archaeological studies in Uppåkra are fairly recent compared to Birka which remains the most well known town and the most studied site and is since 1993 a UNESCO World Heritage Site. I recommend this post about why Birka was special in the eyes of a Norseman at the time.

During the 10th century Birka lost its importance and was abandoned, and around the same time several other towns were founded which survives to this day, including Sigtuna (founded 980, which may be the town that was referenced), Skara (founded 988 according to legend), Lund (archaeological traces from ~990, possibly a relocation from nearby Uppåkra) and Visby on the Baltic island of Gotland (known center of merchandise since 900).

So yes you could claim that the oldest town in present-day Sweden that's still inhabited was founded in 980, but the very first town was founded at least 200 years prior, possibly earlier.