As mentioned there are direct trains from Arlanda airport to Uppsala in about 20 minutes, they are however a bit expensive so another option is the bus. It is a bit slower (40 min) but also cheaper.
Acraven has made a great list, including the treasury in the cathedral is really good. It is a small but excellent museum in the northern tower with some unique pieces of medieval clothes. but there many things I can add. Close to Linnéträdgården is also the Linnéan museum, his old city house. And apart from Linnéträdgården, the old botanical garden, the new botanical garden is also worth a visit. As well as the small museum, or rather exhibition, at the university library.
For older history, Gamla Uppsala (Old Uppsala) was mentioned and I can really recommend the museum. It is small but great, and if you have any question don't hesitate to ask the staff. As far as I know all people working there are either historians or archeologists so they know a lot about the are. In addition there are plenty of rune stones in and around Uppsala so there is a lot of Viking history to see. There are even a couple of rune carvings on the bedrock itself, which is not that common. For even older history there is Hågahögen, the Håga mound (also known as Kung Björns hög, King Björn's mound), a bronze age burial mound in the western outskirts of the town.
For more modern culture, you need to make sure you have time for some fika. It is not only a very important part of life in Sweden, Uppsala is also a great place to explore it. In my opinion the cafés in Uppsala are far better than the ones you'll find in Stockholm. They are an important part of life and studentculture in the town.
If you want to do a day trip somewhere Öregrund is not a bad idea. A charming 19th century town on the coast, about 1:30 by bus from Uppsala. A more odd sight that might be of interest for those that like military history is a field in the southern parts of the town named Pattons hage (Patton's meadow) after the US general that visited the field in 1945.
That is a very short introduction to Uppsala. But if you tell us a bit more about what you are looking for and what you are interested in, I might be able to give you a bit more specific advice. There is also a pretty good article on Uppsala at wikivoyage: https://en.wikivoyage.org/wiki/Uppsala