Ahoy there, Natalie! Welcome aboard, and thank you for sailing by with your question.
Disney Cruise Line has some amazing destinations in
Europe, and this looks like an incredible itinerary. I certainly hope this becomes one of your most magical cruise memories.
Nynäshamn is indeed just under an hour from Stockholm, and there is no complimentary transportation from the port to Stockholm. You will find, however, if you book a
port adventure through Disney Cruise Line, they will include transportation to get you to Stockholm.
Also, right near the port, you'll have access to the Pendeltåg commuter train and a bus, which both take about an hour to get to Stockholm. The prices are really affordable with either option and although subject to change, currently run about $4 each way, per person. The train runs about every 30 minutes, and the shuttle buses run mornings and afternoons. You'll also find taxis at the port and you can hire a private car if you prefer.
I hope this helps you plan for your high seas adventure, Natalie!
Anna