I nearly didn't spot that first ostrich at all. You'd think that a 9ft-high feather duster might stand out a bit on a drab-coloured, scrubby plain, but no; I scanned the foreground, the middle distance and, in desperation, the horizon – still that oversized bird eluded me.
Fortunately, in Gerard I had a driver-guide with the eyesight of a particularly alert hawk, as well as some conveniently hefty background landmarks to aid location.
"OK, so find the skyscraper with the flying saucer on top – the Kenyatta Conference Centre – and follow the line straight down, about 50 yards away. See it?"
And there it was, suddenly obvious against that backdrop of high-rises. I'd just had my first-ever encounter with the planet's biggest bird, gawping back at me in Nairobi National Park, a mere 20-minute drive from the centre of East Africa's largest city.
To say that the juxtaposition of big game and bigger buildings is incongruous would be an understatement. But that's Kenya's capital for you: to paraphrase a well-worn tourism cliché, it's a city of contrasts – just not always in a way that the marketing types would embrace.
Born in the dying moments of the 19th-century, Nairobi was not so much founded as hastily thrown together. A ramshackle cluster of shacks alongside the tracks of the then-new Mombasa-Uganda railway, it was initially blessed with the catchy title "Mile 327", later enjoying a name upgrade borrowed from the Maasai, Ewaso Nyirobi ("place of cool waters").
These days Nairobi has a reputation for crime – and there's no denying the seriousness of the problem, cited in current Foreign Office advice. But after two days, the only larceny I'd encountered was being stung a few shillings over the odds for a painted wooden guineafowl in the market. With one thing and another – including half-price entry visas, down to US$25 (£17) until the end of the year – I'd say Nairobi's a bit of a steal.
Back with Big Bird, now staring at me across the savannah, I did a quick calculation: based on a US$20 park entry fee, my cost-per-game-sighting was already down to well under a buck per beast and falling rapidly. And that within an hour of entering the park.
It might start a mere four miles from the city centre, but Nairobi National Park – Kenya's oldest, gazetted in 1946 – isn't just an urban novelty or open zoo. At 73 square miles (roughly the size of Paris), it's a fully fledged mini-Masai Mara, with plains, woodlands and riverine escarpments – it even hosts its own migration of wildebeest and zebra each July and August.
To explore the park you could hire a car, join one of the half-day tours offered by various operators or even jump on a public minibus tour on Sundays. I was blessed with the expertise and visual acuity of Gerard, our driver from Ngong House, Nairobi's remarkable tree-house hotel. That morning Gerard had packed the cool box with sandwiches and Tusker beers, loaded us in the 4x4, and was now providing us with a running commentary on a fair proportion of the park's 80-plus game species.
Four out of the so-called "Big Five" – lion, leopard, buffalo and rhino – stalk these plains and glades, along with many of the other safari big-hitters: cheetah, antelope, warthog and more than 400 bird species. Though the predators are less easily seen, giraffe and zebra spotted (and striped) the savannah, buffalo watched us warily as we rumbled past, and hippos chuckled in muddy pools.
Gerard was, though, determined to bag us a sighting of a black rhino, rare across the continent but commonly seen here in the park's western woodlands. The difficulty in scouting them was our late entry to the park; our departure from Ngong House had been delayed as we lingered over breakfast. In the heat of the day the rhinos had retreated into the shade, and eventually we did the same.
In a bid to help me tidy up my wildlife tick-list, Gerard whisked us over to the adjacent Daphne Sheldrick Elephant Orphanage. Once again the rhinos were snoozing, but the young elephants, having cooled off with a morning shower, were getting into the World Cup spirit with a five-a-side football game. Understandably a family favourite, with crowds of visitors arriving to watch the gambolling pachyderms enjoying their daily mud bath, the orphanage is also an important conservation centre, having reared many rhinos which have since been dispersed to Tsavo and Nairobi national parks.
On our drive back to Ngong, Gerard detoured through the western suburbs to show us the grand colonial house, now a museum, where Karen Blixen – Danish author of Out of Africa – lived almost a century ago. This year marks the 25th anniversary of the film in which Meryl Streep and Robert Redford played out Blixen's romance against the dramatic Kenyan scenery.
Since then Kenya has faced a succession of challenges, but 2010 could be a vintage year for Nairobi. On top of that visa price reduction, plans have been mooted to allow visa-free access to the city for transit passengers, permitting a tour of Nairobi National Park. A scheme to upgrade the massive Kibera slum hopes to bring electricity, running water and a decent quality of life to many of the city's poorest inhabitants – which might also help improve security for everyone.
And while my specs aren't sufficiently rose-tinted to describe Nairobi as an oil painting, a visit to Kenya's panhandling, hustling capital doesn't have to be an ordeal to be endured between airport arrival and safari or beach. There's plenty to savour in its cultural and – surprisingly – natural attractions.
An alternative 'to-do' list
Be licked by a giraffe
Try some unusual exfoliation, courtesy of the rough, weirdly grey-blue, foot-and-a-half-long oral appendages of the curious Rothchild's giraffes at the AFEW Giraffe Centre (00 254 20 807 0804; giraffecenter.org).
Take your medicine
The quintessential Kenyan cocktail, dawa (Swahili for medicine), mixes vodka with lime and sugar over crushed ice. Sip one at Carnivore, a nyama choma (grill) restaurant renowned for its game meats, though restrictions now proscribe the more exotic animals (00 254 20 602 764; tamarind.co.ke).
See how the other half lives
Take a walking tour of the infamous Kibera slum to learn more about how Nairobi's poorest inhabitants get by (00 254 7 2366 9218; kiberatours.com).
Travel essentials: Nairobi
* Nairobi is served from Heathrow by British Airways (0844 493 0787; ba.com), Kenya Airways (020-8283 1800; kenya-airways.com) and Virgin Atlantic (0844 874 7747; virgin-atlantic.com).
* Ngong House (00 254 7224 34 965; ngonghouse. com). Doubles from Ks19,600 (£161), half board.
* Nairobi National Park: 00 254 20 600 800; kws.org/parks.
* British passport-holders require a visa, which can be obtained on arrival (by air) for £20.
* Kenya Tourism: 020-7367 0931; magicalkenya.com.