There are trains which run normally every 30 mins to and from Arlanda Airport. It's a 18 mins ride. Go to Sky City terminal where you find the train station and take the "SL-pendeln". Cost: 210 SEK one way.
Bus no 801 runs also every 30 mins to and from Arlanda Airport and leaves from both the international and domestic terminals. In Uppsala it stops at the railway station. The ride takes 45 mins. Cost: 90 SEK one way if you buy the ticket in the vendor machines at the airport, or 111 SEK if you buy the ticket at the bus.
Timetable: see the above link, but choose bus 801 under the "Regional buses"
Uppsala Taxi, tel +46-18-10 00 00. The cost is 520 SEK.