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You need tickets that covers your full journey. If you want to travel between Stockholm and Uppsala you need both an SL ticket and a UL ticket. If you combine an unlimited SL ticket with a single ride UL ticket you won't have unlimited trips in the UL area.
When going to Uppsala UL claims that SL sells the UL ticket at manned gates. SL doesn't mention it themselves though. Both the SL app and the UL app also sells the UL ticket (you need UL zone 1,2 and 5). In the UL app there is also a 24h ticket that could be useful.
The airport fee is only paid when departing or arriving at the airport. A ticket for the fee can be bought at the gate at the airport (also when you arrive). It is also sold at manned SL gates and in the SL and UL apps.