Just after dawn on Friday, the climber unfurled a giant banner promoting climate activist group Extinction Rebellion as supporters watched on.
He climbed back down the structure about 10am and was arrested by waiting police who led him away in handcuffs.
The banner he attached to the structure promotes three days of climate rallies set to be held in late May.
The Extinction Rebellion group is known for disruptive, high-profile protests staged around the world.
The group says the May demonstrations will include daily acts of civil disobedience.
Supporters set up tents outside the performing arts centre at Southbank in inner Melbourne to watch the climb on Friday.
The man appeared to attempt to light a flame.
Spokeswoman Jane Morton said the purpose of the demonstration was to draw attention to climate change.
"We are looking at societal collapse, that is what scientists are saying. It's not really on the news. That is why Extinction Rebellion is going to try to sound the alarm," she told the Nine Network.
Tourism, Sport and Major Events minister Steve Dimopoulos visited the arts centre for an unrelated event on Friday and said there were better ways to call for action on climate change.
"Protest is healthy in any democracy but it has to be safe," Mr Dimopoulos told reporters.
"Safe for yourself, but also safe for other Victorians and people who try and then get you out of harm's way (such as) emergency services."
Arts Centre Melbourne has declined to comment on the situation.
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