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The 30-year-old from Auckland, who has been granted interim name suppression, was arrested in Counties Manukau on Tuesday, police told the NZ Herald.
The defendant did not attend his first appearance at the Manukau District Court today because he suffered an injury, his counsel told the courtroom.
He is also undergoing a mental health assessment.
He has been accused of a category three offence, which carries a maximum penalty of seven years in prison.
He has been remanded in custody without entering a plea until his next appearance on May 7.
He has also been charged with harassing a man online "by sending constant social media messages having already sent messages between December 2020 and March 2021", knowing that the harassment is likely to cause the man to "reasonably fear for their safety".
The Prime Minister's office declined to comment.
"As the matter is before the courts, police are not in a position to comment further," police said in a statement.
Threats made against Prime Minister
Prime Minister Jacinda Ardern has been subject to a number of threats during her two terms.
Matthew Burns, 20, tweeted a photo of a gun to the Prime Minister after the Christchurch mosque attacks, adding "you're next".
A man pleaded not guilty to a charge of sending threatening emails to Ardern between 2019 and 2020.
Police accused Michael Christopher Cruickshank of "sending three emails having already sent 89 emails" between October 17 last year and January 19 this year, a charging document viewed by the Herald reads.
The emails, the police allege, were intended to harass the Prime Minister and cause Ardern to "fear for her safety (or) the safety of her family".
A Northern Ireland man was sentenced to community service for sending online death threats to Jacinda Ardern in 2019.
An Australian man who posted online threats to attack a mosque in Newcastle, NSW, and kill Ardern was jailed for 10 months in June 2020.
Cormac Patrick Rothsey pleaded guilty in Newcastle District Court to charges of posting the extremist threats on social media.