She was named to the Úrvalsdeild All-first team for the first half of the 2012-2013 season, at the age of 16, after averaging 15 points and 9 rebounds in the first fourteen games.

[14][15] On 21 February 2019, Keflavík announced that Sara would join the team in March after concluding her season with Canisius.

In September 2019, Sara signed with Leicester Riders of the Women's British Basketball League[17] while also studying for her Master's degree at Loughborough University.

[18] On 15 March 2020, Sara was named MVP after leading the Riders to their third straight WBBL Trophy with a 70-66 victory against Durham Palatinates.

[21] On 25 February 2021, Sara Rún returned to Iceland and signed with Haukar, rejoining with her twin sister Bríet Sif.

[24] In August 2021, Sara Rún signed with CS Phoenix Constanța of the Romanian Liga Națională.

[28] In July 2023, Sara signed with AE Sedis Bàsquet of the Spanish Liga Femenina de Baloncesto.

[37] Sara's twin sister, Bríet Sif Hinriksdóttir,[38][39] is a basketball player in the Úrvalsdeild kvenna and a member of the Icelandic national team.