Going via Stockholm is going to be both the fastest and cheapest option.
Skavsta is really in Nyköping which is a little over an hour south of Stockholm. Arlanda is about 40 minutes north of Stockholm by car. If you were to drive, you'd see the fastest route basically goes through Stockholm.
If it were me, I'd take the flygbus from Skavsta to Stockholm. it goes about every 30 minutes and costs 139 SEK. The Bus goes to Cityterminalen. At Cityterminalen, you can then change to the flygbus to Arlanda; it leaves about every 10 minutes and costs 105 SEK. Some people might suggest the Arlanda Express train, but by the time you walk across the street to Central Station, get to the platform, and wait for the train, this will take you longer than just sticking with the flygbus.
A taxi will be extremely expensive as these airports are not close. According to the Skavsta website it is 1900 SEK (0ver $200) and takes 1.5 hours. Another option is to go to Nyköping train station and take the train from there, but this will take you longer than the bus, cost you more, and you will still have to change trains at Stockholms Central Station
Sorry that there is nothing better to suggest; it is unfortunate that you couldn't fly into Arlanda.