Michael Rogers, dubbed “Trolley Man” online, was filmed on 9 November trying to prevent Hassan Khalif Shire Ali from stabbing two police officers.
The suspect had already killed a cafe owner and injured two other people.
He was later shot by police and died in hospital.
Mr Rogers was called in by police for questioning on Friday.
Australian media reported that the five alleged offences – two counts of burglary, two counts of theft and committing an indictable offence while on bail – had taken place before the 9 November attack.
Mr Rogers has been hailed as a hero for his efforts to prevent Shire Ali from stabbing two police officers.
When tracked down by reporters, the 46-year-old told 7 News: “I threw the trolley straight at him, and I got him. I didn’t quite get him down, though.”
At the time he was only metres from a burning car full of gas cylinders, which the attacker had set alight near Bourke Street, a busy road in the city centre.
An online fundraising campaign for Mr Rogers has raised more than A$140,000 ($103,000; £80.000).
It was set up by Donna Stolzenberg, founder of the charity Melbourne Homeless Collective, to help get him back on his feet.
