Reversible Destiny Lofts

"We have decided not to die," declared architects Shusaku Arakawa and Madeleine Gins in the title of the book they published in 1997, arguing that lopsided, physically challenging spaces would awaken residents' instincts and allow them to live better, longer – even forever. Head out to Tokyo's leafy suburb of Mitaka and see their rainbow-coloured Reversible Destiny Lofts. The nine apartments in this complex have uneven floors and rounded walls, awkward light switches, power sockets in the ceilings, and no cupboards. Some of the units are available for short-term rent, but watch your step!
• 2-2-8 Osawa, Mitaka-shi, +81 4 2226 4966, reversibledestiny.org

St Mary's Cathedral

Kenzo Tange has built many impressive buildings in Tokyo, but St Mary's Cathedral is arguably his finest work. Built in 1964, it is a soaring vision of stainless-steel-clad abstraction. Seen from the air, the apex of its roof forms a cross made of glass. The interior is equally dramatic: a cavern of sloping walls at times lit up in red or blue. All this austerity is counterbalanced by a tall, narrow strip of stained glass behind the altar, which provides the hall with a focus and a crucial, serene touch of warmth.
• 3-15-16 Sekiguchi, Bunkyo-ku, +81 3 3943 2301

Prada flagship store

Tokyo's Omotesando boulevard is a stunning catwalk for high-end contemporary fashion and dramatic architecture. Just when you think the setting can't get any more impressive, the Prada flagship store manages to make its neighbours look like they're trying too hard. Built in 2003 by Herzog and DeMeuron (renowned for the the bird's nest Olympic Stadium in Beijing; their extension to London's Tate Modern art gallery will open in time for this year's Olympics), the Prada store is an irregular construction of green, diamond-shaped glass panels. While many flagship stores intimidate with their austere interiors, Prada's aims to seduce with plush white carpets and rounded plastic walls. It's the epitome of refined but unpretentious allure.
• 5-2-6 Minami-Aoyama, Minato-ku

Nakagin Capsule Tower

Completed in 1972 by Kisho Kurokawa, this eccentric residential/office building – which looks like a stack of washing machines – is an icon of the postwar Metabolist movement. The cramped capsules were originally designed for single businessmen, and each unit contains a built-in shower, bed, TV and phone. These were supposed to be replaced every 25 years, but never were, so the building is now in a sorry state of disrepair. Bought up by an unsentimental hedge fund, it is due for demolition and redevelopment. Get there in the next few months for a last chance to see this hallmark of flawed but visionary idealism.
• 16-10, Ginza 8-chome, Chuo-ku

Fumiko Hayashi Memorial Hall

With the widespread firebombing of Tokyo in 1945 and the postwar reconstruction, there's not a lot of traditional architecture left to see in the city – with one great exception. In 1941, novelist Fumiko Hayashi (1903–1951) built her home here in Edo-era architectural style. Now a museum, this elegant house displays artefacts relating to her life and work, and the tranquil, walled garden where Hayashi wrote her acclaimed novels Ukigumo and Meshi. Spend an hour here and let the clock wind back.
• 2-20-1 Nakai, Shinjuku-ku, +81 3 5996 9207

21_21 Design Sight

In the park behind the high-end Tokyo Midtown shopping centre, two sleek, triangular shards of concrete and glass rise from the ground. 21_21 Design Sight is a signature work by Tadao Ando, and Japan's first design museum. Once you venture beyond the trapezoid lobby, you discover that 70% of the building is underground, but the exhibits still look great thanks to a large light-well and clever spotlighting. Directed by three of Japan's most famous designers – Issey Miyake, Taku Satoh, and Naoto Fukusawa – 21_21 Design Sight's exhibitions present the most artistic examples of international design, from Shiro Kuramata's classic resin "Miss Blanche" chairs to images of Christo's wrapped public monuments.
• Tokyo Midtown, 9-7-6 Akasaka, Minato-ku, +81 3 3475 2121, 2121designsight.jp. Open daily 11am-8pm except Tues

Reiyukai Shakaden Temple

Nothing looms more ominously on the Tokyo skyline than this dark, oblique pyramid clad in black granite and crowned by two rings of gold. Inside, the gloomy, red-felt-lined elevators seem better suited to a David Lynch movie than a temple. Reiyukai Shakaden was built in 1925 for a Buddhist lay sect and houses a large meditation room with a giant Buddha statue. But visitors are welcome, and free Japanese lessons are on offer. The temple also houses a reservoir with 400 tonnes of drinking water for use in the event that Tokyo is struck by a major disaster.
• 1-7-8 Azabudai, Minato-ku, reiyukai.org

Golden Gai Bar District

This compact cluster of more than 200 tiny, ramshackle bars in Shinjuku is the antithesis of planned, efficient architecture. In the 1950s it was a den of black-market trade and vice, but later became a hangout for artists and intellectuals. The chaotic street scenes in Ridley Scott's Bladerunner were apparently inspired by this dingy Tokyo district. Each bar has its own theme, ranging from photography to horse racing. While many of them only serve locals, a few will let you in if you ask nicely. Beware, drinks and accompanying snacks are expensive.
• Kabukicho 1-chome, Shinjuku-ku

Yuzo Saeki's Studio

Yuzo Saeki (1898-1928) was one of Japan's early adopters of western-style oil painting. Late in life, he spent several years in Paris, where he painted self-portraits and landscapes in the Fauvist style. His studio, nestled in a small park in Mejiro, is a very rare find in Tokyo. This intimate, wooden building – reminiscent of an American parish chapel – is flooded with natural light that pours in through a large set of windows at one end, while at the other there's a quirky, irregular row of three doors. Since 2010, the studio has become a museum of Saeki's work.
• 2-4-21 Nakaochiai, Shinjuku-ku, +81 3 5988 0091, free admission. Open Tues-Sun May-Sept 10am-4.30pm, Oct-Apr 10am-4pm

Sonorium

The lopsided white block of Jun Aoki's Sonorium rises up from behind the dark, tiled roofs of a residential street in Eifukucho. The intimate concert hall is a serene place to be on an overcast or rainy day: the sloping walls punctuated by two rectangular windows which become gleaming expanses of bright white. Aoki – known as an artist as much as an architect – has succeeded in creating a building that feels like a three-dimensional work of minimalist geometric abstraction.
• 3-53-16 Izumi, Suginami-ku, +81 3 6768 3000, sonorium.jp

For more information go to the Japan National Tourism Organisation's website: jnto.go.jp/eng

• Ashley Rawlings is the editor of Art Space Tokyo, a guide to the city's most architecturally distinctive museums and galleries