The 40-year-old allegedly made the threats on social media.
Police have also raided a house and storage facility in Sydney, where they seized documents and computer hard drives, the BBC said.
The arrest comes just a week after police said they had foiled a major terror plot which was understood to have targeted central Melbourne.
Seven people were arrested in relation to the planned attack, which had been allegedly due to take place on Christmas Day.
The FCO has not updated its advice since then, but warns that there is a “high threat of terrorism”, and notes that in the wake of the planned terrorist attack, there will be “increased security measures at public events over the Christmas period”.
'Threats' made to Sydney's New Year's Eve celebrations
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