Linköping is located quite far from the nearest airport, so taking a taxi is a very expensive option, bus or train is by far preferable.
From Arlanda Airport:
In order to get to Linköping from Arlanda you first have to get from Arlanda into Stockholm Central station, the most convenient way to do this is to get on the Arlanda Express train which runs between Arlanda and the central station without any stops. It takes about 20-30 minutes and costs is roughly 150-200 SEK. You could also use the Airport transfer buses which takes about 45 minutes and costs about 100 SEK. Both of these options are easy to find at the airport and you can buy your ticket by cash or credit card at the airport, but you can also buy a ticket in advance onwww.arlandaexpress.se or www.flygbussarna.se.
Once you’ve arrived at Stockholm Central Station you need to get a ticket for the trip Stockholm – Linköping. You can choose between train or bus, the train is more expensive and the price varies a lot depending on if you buy it in advance online (www.sj.se) or at the spot, but roughly between 250-500 SEK single trip. The trip takes about 1.5-2 hours but the number of departures is quite limited after 5pm. The bus takes longer, almost 3 hours, but is cheaper (around 200 SEK) and runs more frequently during night. There are two bus companies with whom you can book your ticket, www.swebusexpress.se and www.nettbuss.se. In both cases it is recommended to book a ticket in advance, since the buses often fill up.
From Skavsta Airport:
If you fly with Ryanair you’ll probably arrive at Skavsta Airport, which is situated outside of Norrköping. There are Airport Coaches which runs between Skavsta and Linköping, you can book a ticket on www.flygbussarna.se.