Acheson was born in Tipperary in 1934, daughter of Matt Barlow, Republican activist in the Irish War of Independence. She was educated at the Presentation Convent, Clonmel, the Convent of Mercy, Carlow and the London School of Business Studies. Acheson worked as a company director in the family business before becoming involved in politics.
She was elected to South Tipperary County Council in 1974. She was elected to Dáil Éireann for the Tipperary South constituency at the 1981 general election, but lost her seat at the following February 1982 election.
She is a sister of former Senator Tras Honan.