The two main ways of getting to Uppsala from Arlanda airport to Uppsala is by train or by bus. The train is more frequent and faster (the train takes ~20 minutes, the bus is ~40 minutes), while the bus is cheaper (the bus is 120SEK, so ~11 EUR or USD; the train is typically 197SEK).
The easiest way to purchase tickets is via an android/ios app. One can also buy a ticket directly on the bus by tapping a credit/debit card. Train tickets from Arlanda to Uppsala can be bought from a person at the entrance to the airport train station. Otherwise you can buy train tickets from a ticket machine or at 7-Eleven / Pressbyrån stores.
The bus company is UL, the train companies are SJ, SL and Mälardalstrafik. The SJ app can be used to purchase train tickets for all 3 companies; note that in the UL app you need to buy the "75 minute ticket to/from Arlanda by bus".
Link for train app for android
PS. If you take the bus to Arlanda, do not get off at "AR-terminalen": it may sound like "ARlanda", like the airport, but it's actually a random parking lot20km from the airport.