A decade ago, the streets of Budapest's Eighth District – also known as Józsefváros – were best avoided. Beneath the shadows of the grand aristocratic homes and ornate residences walked prostitutes, drug users and homeless people. Over the past few years, however, the district, which fans eastward from behind the National Museum, has changed its reputation: easygoing cafes and art spaces are filling the gaps in an area laden with faded glamour. This is the place to head if you want to swap traditional sites and rowdy bars in favour of crumbling balconies and local haunts.

One block behind the National Museum (Múzeum körút 14-16), which is packed with medieval swords and cold-war posters – and past some of the looming mansions that have led to the inner Eighth District being dubbed "the palace quarter" – is the main hub of the district's nascent bohemian scene: a row of shops, cafes and restaurants running along Krúdy Gyula street and around the sides of peaceful Mikszáth Kálmán square.

The first place you'll see will undoubtedly be the Zappa Cafe (Mikszáth Kálmán tér 2), a corner bar whose interior is dominated by a striking mural by French artist Jean-Michel Verret, depicting the view from his New York apartment. The painting is a relic from the venue's previous incarnation, as Tilos az Á – an edgy underground music venue that opened in 1989 – and the cafe is now named after Tilos's most prominent performer: Frank Zappa. It still plays host to the local alternative community – just round the corner is the entrance to its basement club, Trafik, an atmospheric cellar where you can immerse yourself in the Hungarian punk scene.

Next door, also with outdoor seating spilling into the square, is the Lumen Gallery and Cafe (Mikszáth Kálmán square 2), which holds regular exhibitions of contemporary art, as well as organising evening gigs in the cafe. If you're there during the day, you'll find a light, modern space where you can have a coffee alongside busy-looking creative types. In the evening, it's a hip bar where you can enjoy a pint of the excellent local IPA.

Further along the street there are a couple of shops well worth dropping into. The first, Ethno Sound (Krúdy Gyula utca 7), looks at first glance like a hippy gift shop. In its basement, however, is an unbelievably comprehensive collection of percussion instruments from all over the world. Most people find it impossible to leave without having tapped, flicked or shaken everything in sight.

A couple of doors down is Iguana (Krúdy Gyula street 9) – a den-like vintage shop filled with retro garments of twice the quality and at half the price you'd find in London.

Those looking for hearty Hungarian grub should make a beeline for Darsham Udvar Étterem (Krúdy Gyula street 7), a place where you can fill up on meat and dumplings and wash it down with a beer for around a tenner a head. But it's worth venturing into the outer Eighth to find one of the district's nicest places to eat: Csiga Cafe (Vásár utca 2). The laid-back spot, with friendly staff and a cosy interior, serves delicious, well-presented dishes, such as traditional borscht (£1.30) and duck leg with pumpkin purée (£5.50).

Once your stomach's lined, it's time to hit some of the bars. The area will seem a world away from the stag parties stumbling around the seventh district and its famous "ruin pubs" in derelict houses. The Eighth does have a couple of "ruin" bars of its own: the grittier (and quieter) than usual Gondozó Kert (Vajdahunyad street 4), in a courtyard opposite a homeless shelter, and Corvinteto (Blaha Lujza tér 1-2), a former supermarket with a huge roof terrace, which keeps electronic beats going until 6am Wednesdays to Saturdays. Or buzz yourself in through the discreet entrance for Müszi (Blaha Lujza tér 1), a huge warehouse arts space, which has exhibitions, a cafe and a barbershop to enjoy in the day but still manages to host rowdy club nights with a shabby house-party feel. Wherever you end up, as long as you're not wearing a shirt, you won't feel like a tourist.

Local's tip

Nelli Virágh, manager at Csiga Cafe: "Another really nice pub is Macska (Berkocsis street 23), which is also a vegetarian restaurant. (The word macska means cat.) They have lots of events with music – sometimes famous bands play there and sometimes it's just friends."

Where to stay

Almost a reason in itself to visit the Eighth District, Brody House is a low-key boutique hotel and undoubtedly one of the most chic places to stay in the city. This 19th-century building, with a ludicrously vast stairwell, a distressed interior and vintage furniture that fits seamlessly with its crumbling palatial surroundings, is right at the tip of the district, facing the National Museum. This makes it the perfect location from which to take a leisurely 15-minute stroll across the Danube for a soak in the majestic Gellért Baths. Later, once you're done with the hotel's honesty bar, guests can head to Brody Studios – the hotel group's recently opened, private members' club near Oktogon (a five-minute taxi ride away), where you can enjoy a cocktail and a dance with the city's classier set.
Bródy Sándor street 10, +36 1 266 1211, brodyhouse.com. Doubles from £57, room only

• Flights with Wizz Air and accommodation were provided by Hungarian Tourism (gotohungary.com). Wizz Air flights from Luton to Budapest cost from £30 one way