Crossing Iceland by foot is one of the most amazing experiences. This route is from Akureyri to Skogar - the most popular. It is possible to arrive at Akureyri by plan from Reykjavik. It shares some part with the other N-S long trail on the FKT site and also follows the famous Laugavegur-trail. The start is by Akureyri airport and the end at the seaside outside Skogar following the river.
Here is the best route documentation:
https://www.reddit.com/r/VisitingIceland/comments/ut3044/trip_report_iceland_coast_to_coast_akureyri_to/