THE grey steel arch of the Sydney Harbour Bridge loomed above me, an awe-inspiring 400 feet above the sea.

“It will be sensational,” promised my guide as we prepared to scale what was once the Australian city’s tallest structure.

Easy for her to say.

Athletic and enthusiastic, Chrissy Le Breton has made about 1,000 climbs of the iconic bridge. As a first-time visitor to Australia, ready for must-do experiences, I suppressed a fear of heights to take her word for it.

I’d had plenty of time to contemplate this dizzying ascent – from 32 floors up – as the sitting-room of my lofty harbourside hotel, the five-star Shangri-La, overlooked the 75-year-old bridge and the equally iconic Sydney Opera House.

After passing the mandatory pre-climb breath test, then being zipped into a blue-and-grey climbing suit, linked to Chrissy by radio and attached to a reassuring safety cable, we set off on a vertiginous 3½-hour tour of the world’s largest steel-arch bridge.

The Discovery Climb – one of two routes available through operator BridgeClimb Sydney – takes visitors up 1,090 steep steel stairs and narrow-mesh catwalks over and through the heart of the giant structure. From a tiny platform at the summit, harbour craft and traffic on the road deck far below look like toys.

Buffeted by a warm breeze high above the glittering blue water, my trepidation was overcome by an incredible sense of achievement – and spectacular views of the harbour’s 150-mile shoreline and the city’s sprawl to inland mountains.

We marvelled at the courage and expertise of men who almost 80 years ago, without safety equipment, assembled the 39,200-tonne steel curve which was locked in place by 18-foot, hand-hammered rivets.

The spectacular harbour, busy with ferries and tour boats, is dominated by the bridge and the dramatic Opera House – now a World Heritage Site.

The Opera House, visited by 4m people a year, is by no means elitist. Each year it stages around 2,500 performances and events, from a bargain £12.50 per ticket.

You can even enjoy a performance for the price of a high tea there, with an opera singer giving a live performance among the tables.

My brief time in Sydney was spent mainly in The Rocks area – developed after British First Fleet convict ships anchored in 1788 off a small peninsula beneath where the bridge now stands.

Now overlooked and dwarfed by canyons of gleaming skyscraper hotels and commercial buildings, The Rocks is a glimpse of how Sydney used to be – albeit now transformed into cafes, boutiques and restaurants – and where traditional hotels still offer a welcome beer and sanctuary from the heat.

Sydney has a vibrant arts and culture scene, and my visit included the hottest ticket in town: Australia’s stage and screen icon, Cate Blanchett, was starring in a two-part War of The Roses, an adaptation of eight Shakespeare plays for the 2009 Sydney Festival.

Blanchett, gold-crowned and clad in white trousers and shirt while gold foil rained for an hour onto the vast, bare black stage, mesmerised as King Richard II in the powerful, bleak and bloody four-hour performance I saw.

As a contrast, I sampled the sun, sand and sea of Sydney’s beach culture, surfing at the world-famous Bondi Beach, a short drive from the centre.

Coated in high-factor sun block against scorching heat, I strolled along a stunning coastal path after a pavement-table breakfast at Swell Restaurant, in genteel beach-side Bronte, where stars including Australians Nicole Kidman and Hugh Jackman are said to sample the laid-back atmosphere.

Fortunately, the legendary surf at Bondi was kind, as “Big Wave” Dave Hannagan, of the Let’s Go Surfing school, pledged to have me – a total novice – standing on a surfboard after an hour’s lesson.

After some expert tuition and some undignified duckings, he did – for about three seconds.

Melbourne, just an hour’s flight away, was to me a city of secrets.