A few thoughts:
If you haven't already then look up your itinerary on the IcelandAir web site and make sure that the details are as you expect them to be. Fill in any information, such as passport details, that may be missing. They may want a mobile number and an emergency contact.
Check-in online but don't worry too much if you can't get boarding passes. The airline will want to see your passport at the airport and may want to force you to visit the check-in desk even if you aren't checking a bag.
Allow plenty of time before the flight because nothing is worse than starting the trip with waiting in long lines followed by a mad dash to the gate.
Once you have done what you can, relax and go with the flow (or lack thereof) as there is no point in fighting it! Have a good trip.