Arnstad was born in Stjørdal. She held various jobs and offices before entering politics on a national level in 1993. That year, she was elected to the Storting and sat as a representative for the Centre Party from 1993 to 1997. Following the 1997 election, she took a seat in Kjell Magne Bondevik's first Cabinet as Minister of Petroleum and Energy from 17 October 1997 to 17 March 2000, when the government resigned over the issue of gas power stations. In 2001 she was once again elected to the Storting, where she sat until 2005.