Staur holds a MA degree (History and Literature) from the University of Copenhagen.

He was State Secretary in the Ministry of Foreign Affairs from 2001 to 2007, with responsibility for the implementation of the Danish International Development Agency (DANIDA).

[3] As State Secretary, he was also in charge of Denmark's cooperation with the United Nations system and the World Bank.

He also served as vice-chair of the council of the International Organization for Migration (IOM) from 2016 to 2018; and was a member of the board of the Global Fund to Fight AIDS, Tuberculosis and Malaria (GFATM) from 2005 to 2007 and again from 2016 to 2018.

In 2022, Staur became Denmark's candidate to succeed Susanna Moorehead as chair of the Organisation for Economic Co-operation and Development’s Development Assistance Committee;[4] in the final vote, he won over Nikolai Astrup, who won the support of 13 members in the ballot compared to Staur’s 16.