Maps and travel hints:
If coming form Arlanda Airport or Stockholm (City) Central Station one of the easiest ways to get to Karolinska is to go by commuter train “pendeltåg” (NOT subway “tunnelbana”) to Stockholm Odenplan Station and exit north towards Vanadisvägen and then walk 850m to the hospital main enternace at Eugeniavägen 3, see map for walking https://goo.gl/maps/k3VjYS1djMt35fxD8
Another way is to go by subway “tunnelbana” (from e.g. T-Centralen) to S:t Eriksplan (exit Torsgatan Karlbergsvägen) and then buses from there two stops to Karolinska sjukhuset Eugeniavägen
Map of all trains and subways: https://mitt.sl.se/ficktid/karta/vinter/SL_Spårtrafik.pdf
Note that commuter train 40 goes between Arlanda and Odenplan. There are also airport trains and busses from Stockholm central station to Arlanda airport.