Does the Hotel Wellington have an airport shuttle?
What does the hotel website say?
Arlanda airport is 45 km away. Normal way would be to take the Arlanda Express train to Stockholm Central station, about 1 1/2 km from the hotel. The 54 bus will get you within a couple of blocks, or get a taxi.
We are going on the Scandinavian tour June 2-15 and I have just purchased bus tickets on line from the airport and then metro on to hotel. The directions were on the Wellington information in the tour packet. Nothing was said about a shuttle in the tour information. Hope this helps.
I found the max taxi rate in the city zone is about 650 K.
The train is about 220 K per person.
Not sure what the bus costs?
Very few hotels in Europe have airport shuttles, unless they are right in the vicinity of an airport (even then, not all do). So, unless it is specifically listed on a hotel's website, assume they don't.
There is no need for hotels to have airport shuttles when there is excellent public transport available.