Police say shots were fired in the terminal's check-in area on Sunday afternoon local time. No-one was reported injured.
The man was quickly detained, and a firearm recovered. His motives remain unclear.
The capital's airport was evacuated as a precaution.
Australian media report that the gunman fired approximately five shots. Bullet holes were later seen in the glass windows of the terminal.
Footage has emerged on social media purportedly showing a man being arrested by police officers at the airport.
Witnesses described chaos and confusion at the airport's check-in area as security guards urged people to leave.
One eyewitness has told the BBC she hid under a desk in one of the airport lounges.
Alison, another witness, said she had just put her bags onto the security check belt when the shooting started.
"We were in security and heard the first gunshots. I turned around and there was a man standing with a pistol, like a small one, facing out towards the car drop-off," she was quoted as saying by the Sydney Morning Herald newspaper.
Lily Thomson, a reporter for public broadcaster ABC, said she heard the shots and realised something was wrong when people started screaming "run".
"I saw this grandma with a baby and the fear in her face and I knew it was something serious.
"So we all ran and I stayed with that grandma and her baby and hid behind an information desk. We stayed there for a couple of minutes until security told us to evacuate out to the car park.
"Everyone was hiding behind chairs and people were running. It was terrifying," Ms Thomson said.
Some passengers were stuck on board the aircraft on the tarmac as police officers swept the terminal. The passengers were later allowed to disembark.
Some flights have been cancelled or rescheduled following the shooting.