The 40-year-old man was arrested at Sydney Airport on Thursday after arriving on a plane from London.
The alleged threats were made on social media, said Acting Deputy Commissioner Frank Mennilli.
Police also raided a house and a storage facility in Sydney, seizing documents and computer hard drives.
There was no remaining threat to the community, police said. However, they would not provide details of the alleged threats while the matter was before the court.
The arrest comes a week after police said they foiled a major terror attackplanned for Melbourne on Christmas Day.
Mr Mennilli said the man did not have links to any groups.
"He has been charged with a crimes act offence, not a terrorist offence," he added.
More than one million people are expected to attend Sydney's annual New Year's Eve fireworks party on the harbour.
Officers from the NSW Police Force Terrorism Investigation Squad made the arrest after a tip-off from the public.
Deputy Commissioner Catherine Burn said police believed it was an isolated incident.
"As a consequence, we are confident that there are no current or specific threats to New Year's Eve and, while we ask people to remain vigilant, people should enjoy the celebrations," she said in a statement.
"The community should be also reassured that we will take swift action to ensure the ongoing safety of the public."