Bus no. 801 from Arlanda airport to Uppsala C bus/railway station stops at all terminals (timetable attached). It runs at least twice an hour. A one-way ticket costs 96 SEK. The ride will take about 40 minutes. Tickets can be purchased at the information counter in every terminal (ask there for the bus stop location or look at the info table at the exit) or paid by credit card at the bus (117 SEK).
SL's commuter train to Uppsala departs from Arlanda Central Station in SkyCity, the area between the terminals 4 and 5. It runs twice an hour, and a ticket costs 216 SEK. The train will not save you much time because of longer walking distances.
Do NOT use the Arlanda Express. You will end up in Stockholm!
A taxi from Arlanda to Uppsala should be booked at least 40 min in advance, e.g., Taxi 100 000, tel. +46 18 100 000. Up to four persons pay 520 SEK for the ride, which will take about 30 minutes. Beware of taxi drivers, there is some cheating.