In search of... Axel Munthe in Capri

How does a serious Swedish doctor fit into an island famed for its licentiousness?

If, like so many holidaymakers bound for the Amalfi coast in southern Italy, you've dismissed the island of Capri as overcrowded, expensive and eclipsed in the fashion stakes by Sardinia's Costa Smeralda, think again.

Certainly, thousands of tourists still flock here from May to August in search of beaches, designer shops and overpriced cappuccinos. But the cunning traveller steers clear of Capri in the summer months. From October to April, this rocky, pine-cloaked chip of limestone, barely five miles square, is restored to its natural beauty, and its many cultural sites, including Axel Munthe's magnificent Villa San Michele, are virtually empty.

You go to Capri for culture? I thought it was all cavorting in grottoes and lounging around in the sun

A fair bit of that goes on. Capri town is a fashion victim's paradise, abuzz with designer shops and cafés; as four women in our 30s on a bit of a spree, I can't say we suffered too much. But culture is never far away. Hedonists have been indulging themselves here since Roman times.

And what difference does that make?

Two millennia ago, Emperor Tiberius built 12 villas on Capri, where he indulged in sexual shenanigans with the island's beautiful young boys and girls. Consequently, the place is littered with ancient ruins. Most impressive is the Villa Jovis. On the north-east cliffs, it is a muscle-crunching, half-hour hike out of Capri town, but the sweep of crumbling terraces and Romanesque arches makes it worth the effort. Keen classicists should also check out the Damecuta Archaeological Park in the west of the island.

So where does Mr Munthe fit in?

Munthe was a Swedish doctor, based in Paris and Rome in the late 19th century. Some say he was a charlatan, others a hero – he did sterling work in the 1884 cholera epidemic in Naples – but he is most famous for building the Villa San Michele, just outside Anacapri, the island's only other town. Munthe's vision was quite obsessive. In his autobiography, The Story of San Michele, he wrote that the villa would contain "nothing superfluous, nothing unbeautiful, no bric-à-brac, no trinkets", and San Michele is famous for its unique architectural style, extraordinary array of treasures and beautiful gardens.

Sounds interesting. Where is it?

The villa is a short westerly walk out of Anacapri town at the end of a street of shops selling typical Caprese fare. My friends were so exhausted by buying all things lemony – from limoncello liqueur to lemon-scented perfumes with alluring names such as Capri Nights – they turned up their noses at Villa San Michele and set off back to Capri.

To be fair, from the outside, the villa looks like just another Mediterranean building, all blinding white stone and barred windows. Only the scattering of antiquities and the Norman arched doorway, bordered by a marble frieze and mosaic, suggest that the interior might be more interesting.

Didn't I read somewhere that some of the pieces inside are fakes?

It's true that some local dealers ripped Munthe off with copies, but my guidebook, bought inside the villa, pointed out which pieces weren't original, and it didn't spoil the overall effect. The word eclecticism could have been coined for this place. Each room contains exquisite objects, from antiquities – the oldest is an Etruscan head from 450BC – to Christian iconography and Renaissance and Baroque furniture. I was especially taken with the 15th-century wrought-iron Sicilian bed and the 17th-century copper saucepans.

But the high point was roaming around the garden, along pergola-shaded walkways and terraces bright with flowers. Magnificent trees include a cedar of Lebanon and an avenue of cypresses planted with cuttings from the Villa d'Este in Rome. To the west stretches the Bay of Naples, with Vesuvius's misty hump in the background. Thanks to the unfashionable time of year, I had the place almost to myself, and its tranquil beauty was bewitching.

Just above the remains of yet another Tiberian villa, I came upon San Michele's most handsome resident: an Etruscan sphinx perched on a parapet on the very edge of the cliff. This slab of red, speckled granite – like a huge, bony-haunched marmalade cat – is some 3,200 years old. Rather creepily, her seaward gaze makes it impossible to see her face, and Munthe's mystical account of her origins in The Story of San Michele only adds to the spookiness. An old wives' tale tells you to stroke its sides and make a wish; mine was to come back in spring when the wild flowers will be at their best.

That's enough culture. Let's have a bit of that hedonism you mentioned

No problem. Capri's coastline is peppered with grottoes. Most famous is the Grotta Azzurra, renowned for the extraordinary blueness of its water. You can get there by boat or bus from the main harbour at Marina Grande. We treated ourselves to a taxi, which just happened to be a convertible Golf driven by a Leonardo DiCaprio lookalike. He swept us west to Anacapri, around treacherous bends and a rocky landscape lush with flowers, vines and citrus groves. Thanks to the benign climate, the island is particularly blessed by nature – there are more than 840 species of flora – and it's a botanist's paradise in spring. At Anacapri, a glamorous Mediterranean sprawl of trinket shops, bars and dazzlingly white houses, we caught the shuttle bus to the grotto.

On first sight, it was rather alarming: a shoal of bobbing rowing boats, filled with bewildered tourists and noisy Italian boatmen, swirling around a black tunnel mouth. In fact, once we'd clambered into the boat it was great fun – even the bit where you have to fling yourself on to the deck while the oarsman tugs you through the tunnel on a length of chain.

And once you're inside it is truly, madly, deeply blue. The weird phosphorescent brightness is apparently due to the angle of the sunlight, but I understood why a century ago locals said the grotto was haunted by witches; the vivid colour in such a dark place felt distinctly uncanny.

All this loveliness works up an appetite. Where can I get some refreshment?

The easy option is to grab a pizza in Piazza Vittoria, the buzzy square in the centre of Anacapri. However, I took the bus back to Capri town to meet up with my friends. They were even more ravenous than I, having just hiked all the way from Anacapri via the Phoenician Steps, a two-mile-long rollercoaster of a staircase, built, in fact, by the ancient Greeks. Despite aching limbs, they insisted the stunning scenery made it worth the effort.

Feeling virtuous, we treated ourselves to tea and cake in the wood-panelled bar of the Hotel Quisisana; think five stars, two pools and about an acre of gilt and marble. Call me Eurotrash, but I'd stay there with pleasure. Those looking for old-fashioned Caprese cuisine should try Da Gemma. Opened in the 1940s, this snug haven is found under the arches in Via Madre Serafina and was a favourite eaterie of on-off resident Graham Greene.

So where did you stay, if not the Quisisana?

On the mainland in the Grand Hotel Excelsior Vittoria in Sorrento. This is an old-fashioned, Belle Epoque gem with fresco-dripping ceilings, a pool encircled by lemon trees, and superb views over the Bay of Naples, plus the simpatico service you find only in family-run places. As it happens, in the 1920s Queen Victoria of Sweden stayed here regularly when she came to visit her favourite doctor, the aforementioned Mr Munthe, a relationship which some hint transgressed professional boundaries.

So how do I get there?

There are regular ferries and hydrofoils to Capri from both Sorrento and Naples. Our trip was arranged by Exclusive Italy (01892 619650; www.exclusiveworldwide.com). A package including return flights from Gatwick to Naples, transfers and three nights at the Grand Hotel Excelsior Vittoria (www.exvitt.it) costs from £574 per person, based on two sharing. Go (0870 607 6543; www.go-fly.com) flies daily to Naples from £80 return. Doubles at the Excelsior are from £185, with discounts until 1 April. For more information on Capri, visit www.capri.net.