Best Travel advice from airport to Wellington hotel in Stockholm

Define best please. And which airport?

But, assuming Arlanda: Either train or coach from the airport to the central station and then metro to Östermalmstorg, use exit Sibyllegatan and walk the last 200 metres to the hotel.

From Hotel website:

Find the hotel
Train
Train to Stockholm C,
change there to metro or taxi.
Subway
Red line towards Ropsten or Mörby C,
get off at Östermalmstorg.
From Arlanda Airport
Arlanda Express or Flygbuss to Stockholm C,
change here to metro or taxi.
Taxi
We recommend Taxi Stockholm,
phone +46 8 15 00 00.

The advice above about the Arlanda Express and the Metro is good. It was pretty easy when I was in your situation several years ago. However, depending on your budget, how much baggage you have, whether you've been flying all night and are a bit tired, I might opt for a taxi.

If you want to opt for a taxi, my suggestion would be to take the train (or coach) to Stockholm and then a taxi to the hotel, instead of taking a taxi from the airport. The trip would be a tad more complicated, but cheaper and probably faster.