Unnsteinn Árnason began his musical career at the Mosfellsbær Music School and from there completed an intermediate exam on classical guitar under the guidance of Kristinn Árnason. In 2008 he began his studies at the Reykjavík Academy of Singing and Vocal Arts, first under the guidance of Alexander Ashworth and then Garðar Thór Corter, Viðar Gunnarsson and pianist Krystyna Cortes. In 2012 Unnsteinn received a grant from the Vilhjálmur Vilhjálmsson Cultural Fund and in 2013 he won the competition "Young Soloists" and sang a concert with the Iceland Symphony Orchestra in Eldborg, Harpa.

From 2013 to 2017 Unnsteinn continued his vocal studies at the University of Music and Performing Arts, Vienna under the guidance of Karlheinz Hanser. In 2016 Unnsteinn was chosen to take part in the "Young singers project" at one of the most important opera festivals in Europe, the Salbzurger Festspiele. After graduation, Unnsteinn was part of the ensamble at the Tiroler Landestheater in Innsbruck, where he sang over a dozen roles on stage and worked there until 2021.

Unnsteinn was chosen as the most promising young singer at the Austrian Music Awards 2019 for his portrayal of Mr. Kofner in Menotti's opera Der Konsul. In the season 2021-2022 , Unnsteinn sang Sarastro in Mozart's Magic Flute in Innsbruck and the role of Polyphem in the new opera Die Odyssee at the Zurich Opera. In the summer of 2022, Unnsteinn sang at the Bregenzer Festspiele, the roles of Comissario in Puccini's Madame Butterfly and Walinoff/Governatore in Siberia by Umberto Giordano.

Unnsteinn is one of the founders of the Icelandic Chamber Opera (Kammeróperan), which is a newly founded opera company in Iceland with the aim of increasing the availability of smaller opera productions in Iceland. He played the role of Guglielmo in Mozart's Cosi fan tutte in the first production of the Chamber Opera in October 2022 in Iðnó and then February 2023 in Gamla Bíó. In March 2023, he sang the role of Comissario in the Icelandic Opera's production of Madame Butterfly.

He will play the role of Figaro in the Chamber Opera's new production of The Marriage of Figaro at the Reykjavík City Theatre, which will be performed in February 2025. In March 2025 he will play the role of Benjamin in the premiere of the opera BRÍM by Adolf Smári Unnarsson and Friðrik Margrétar-Guðmundsson. Unnsteinn will play the role of Odin in the premiere of the opera Ragnarok by Helgi Rafn Ingvarsson in October 2025.