David McWhinnie, 49, hurled abusive remarks at the Labour politician as she was canvassing in the East End of Glasgow, the area she represents at Westminster.
Mrs Curran, 55, was called a "slut" by McWhinnie.
Police officers were passing Amulree Street, where the incident took place, and were alerted but McWhinnie continued to shout and swear.
At Glasgow Sheriff Court, McWhinnie pleaded guilty to behaving in a way that would cause a reasonable person to suffer fear or alarm by shouting, swearing and uttering offensive remarks towards Mrs Curran, on November 23 last year.
He also admitted placing others in a state of fear and alarm by making violent threats in Culrain Street, Shettleston, on December 14, 2013 and on December 18, stealing a bicycle from Hill Street in Glasgow city centre.
Harvie Diamond, defending, told the court that McWhinnie has converted to Islam and complained he is having a hard time from other prisoners because of his religion.
Sentence was deferred until next month for reports. McWhinnie was remanded in custody.
Mrs Curran, Shadow Secretary of State for Scotland, has been an MP for Glasgow East since 2010.
She was an MSP for Baillieston from 1999 until 2011 and held several posts in the Scottish Executive.