So I have it all planned out - I will be interested in seeing how well
my plan will resemble what actually happens once I begin this trip. So I am presenting this as how one can go from the Stockholm-Arlanda Airport to the island of Utö, though on a theoretical level, based on timetables and facts that are readily available on the internet.
My trip begins with the arrival of my flight (SAS) at the Stockholm-Arlanda Airport at 12:05 in the afternoon.
1. Stockholm-Arlanda Airport to Stockholm Centralstationen
There is a high-speed train called the Arlanda Express (website) that goes directly between the airport and Stockholm's central train station in only 20 minutes. Trains leave the airport 4 to 6 times per hour, and so it will not be difficult to find one. According to the timetables available on the website, there is a train that leaves the airport on the hour at :05, :20, :35, :50 between 5AM and 10PM during the summer (outside of those times, there are fewer trains. Therefore, I will take the 12:35 train, which will give me 30 minutes to purchase my ticket and get to the train platform.
If possible, however, it would be best to get on the 12:20 train. By doing this, I will be able to have more time to get from the bus at Årsta Brygga to the ferry (one hour as opposed to one minute).
The
Arlanda Express platform is beneath the airport, and so you need to
take one of the 4 escalators that go down to it. There is one in
Terminals 2 and 3, one in Terminal 4, and two in Terminal 5 (see a pdf map of the airport). Right next to the escalators is a yellow ticket machine where you can purchase the train tickets. An adult ticket is 220 SEK at the time of the writing of this entry. I recommend purchasing an e-ticket beforehand however (see comments below).
2. Going from Stockholm Centralstationen to Västerhaninge centrum
The
Arlanda Express will arrive at Stockholm's Central station at 12:55,
either platform 1 or 2 (as seen in the photo to the left). I figure that this will be an ideal time to have a quick lunch. I would like to not hang around too long, so that I can get to Utö as soon as possible, and so it is likely that I will probably just buy a bottle of water and eat a lunch that I made prior to leaving on the trip.
At any rate, it will be necessary for me to go to the pendeltåg (local commuter train, see the image below), which are on platforms 13-16.
The train that I need to get on is one that will take me to
Västerhaninge Centrum. This will be the J35 train, which runs
North-South from Bålsta to Nynäshamn. According to the time schedule, I
will want to get on the 13:37 train. It appears that I can simply purchase a "1-timmes zonbiljett" (a one-hour travel pass) at any of the Biljettautomater (ticket machines). From what I have read, it appears that for the pendeltåg, you purchase tickets by time periods, rather than on a voyage-by-voyage basis.
UPDATE: once you arrive at the station, you will walk through the glass doors and follow the signs to the Pendeltåg. Then, once inside, you will see to your right the Pendeltåg entrance which shows the arrival and departure times, and the various tracks where your train leaves from. Note that the train to Västerhaninge does indeed leave from track 14:
You will do down some stairs, to a lower level that gives you access to the various pendeltåg tracks. There is a ticket counter there right by the turnstiles that let you onto the pendeltåg tracks, so you can just buy your tickets there.The trip from Stockholm to Västerhaninge will take only 30 minutes (calculate times). Therefore, I believe that I will arrive at Västerhaninge Centrum at precicely 14:07.
3. Västerhaninge Centrum to Årsta Havsbad
I will now need to find bus number 846, which goes towards Årsta (Karlslundsplan). As you can see, the busses are right next to the train platform:
The first bus that will arrive around the time I show up is at 14:28. According to the bus schedule, the trip will take 16 minutes and I will thus arrive in Årsta brygga at approximately 14:44.Here is the bus platform on-location:
UPDATE: This part of the trip did not go accoring to plan. Upon looking at the posting for the bus 846 schedule, it posted two different destinations, neither of which said "Årsta Brygga". And the times seemed way off.This was not an issue, however, because about 10 meters away is a taxi station. I therefore just got on a cab and was there in just a few minutes!
Årsta Brygga is where the ferry departs:
Departure point:
For information, the bus stops right in front of that sign, so if you do take the bus, you will not have any problem finding the ferry!
4. Årsta Havsbad to Spränga, Utö by the Waxholmsbolaget ferry
I will have exactly one minute to get from the bus stop to the ferry!
There are daily departures via the Waxholmsbolaget (website) ferries that go from Årsta Havsbad to the island of Utö.
The ferry that I want to get on is called the "Silverpilen" (silver arrow, in English), and it is number #2109, which departs at 14:45 (note: the photo below is just to show the ferry, not the location, which is on Utö).

I chose this one because it stops at the Spränga harbor in Utö, which is near to where the stuga is that I will be staying at. The trip is scheduled to take 45 minues, which means that I will be arriving at Utö at 15:25 at Spränga.
UPDATE: The ferry trip was very nice, and I took the time to relax and have some coffee. You purchase your tickets for the ferry right there on the boat after it leaves the docks. You can buy snacks on the boat, and even some sandwiches and a map of Utö.
Arrival:
View looking out from shore:
5. Conclusion
In all, the journey from Stockholm's Arlanda airport to Utö should take about 3 hours and cost about 50 euros in total. This is quite reasonable, and I am curious to see how this plays out in reality...
Here is three versions of the same trip (starting from Stockholm Central Station) as defined by the SL trip planner:
Leaving at 12:52:
Ta pendeltåg 35 från Stockholms central mot Västerhaninge.
Det går från Stockholms central 12:52
och kommer till Västerhaninge 13:22.Vid Västerhaninge station byter du till buss 846 mot Årsta (Karlslundsplan).
Den går från Västerhaninge station 13:28.
Du är framme vid Årsta brygga ca 13:44.Leaving at 13:22:
Ta pendeltåg 35 från Stockholms central mot Västerhaninge.
Det går från Stockholms central 13:22
och kommer till Västerhaninge 13:52.Vid Västerhaninge station byter du till buss 846 mot Årsta (Karlslundsplan).
Den går från Västerhaninge station 13:58.
Du är framme vid Årsta brygga ca 14:14.Leaving at 13:52:
Ta pendeltåg 35 från Stockholms central mot Västerhaninge.
Det går från Stockholms central 13:52
och kommer till Västerhaninge 14:22.Vid Västerhaninge station byter du till buss 846 mot Årsta (Karlslundsplan).
Den går från Västerhaninge station 14:28.
Du är framme vid Årsta brygga ca 14:44.
UPDATE: All went according to plan (with the exception of taking the taxi instead of the bus), and I arrived right on time.

TIP: I just discovered that you can purchase your Arlanda Express ticket online in advance (https://www.arlandaexpress.com/eticket/start.aspx). Then, you can just go right onto the train, without needing to purchase a ticket. In this case, your ticket is your credit card, though you also need to present a photo ID.