Peter Nunn, 33, targeted opposition Labour MP Stella Creasy in July and August 2013 after she backed a campaign to put author Jane Austen on the banknote of 10 British-pound.
Both Creasy and feminist Caroline Criado-Perez, who led the campaign, said they were terrified after being subjected to widespread Twitter abuse for their views.
Nunn, a father-of-one from Bristol in western England, had denied a charge of sending indecent, obscene or menacing messages, but was found guilty earlier this month.
The part-time delivery driver also wrote: "If you can't threaten to rape a celebrity, what is the point in having them?"
Sentencing Nunn at City of London magistrates' court, judge Elizabeth Roscoe said his behaviour was "entirely unacceptable", and imposed a restraining order banning him from contact with either Creasy or Criado-Perez.
Nunn had claimed he sent the messages to exercise his right to freedom of speech and to "satirise" the issue of online trolling, and declared himself in his trial to be a feminist.
Criado-Perez also welcomed the jail term but said she thought Nunn should have been charged with stalking.
"This man made me fear for my life as no-one ever has before. I felt he was a clear and present threat to me," she wrote in a blog posting.
"He made me scared to go outside, to appear in public. He stopped me being able to sleep; being able to work. He seemed obsessed enough to carry out his threats."
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