This musical boasts an impeccable pedigree. It is based on the 1960 Billy Wilder film The Apartment. It has a book by Neil Simon, marks the only excursion of composer Burt Bacharach and lyricist Hal David on to Broadway, and had a long New York run in 1968. Yet, as an archetypal product of the 60s, it feels faintly anachronistic, both in style and content, in today’s world.

Part of the problem lies with the story. The Wilder movie offers an acerbic study of a corrupt society in which a young insurance clerk, CC Baxter, gains promotion by allowing his superiors to use his apartment for their extramarital affairs. The bitter twist comes when he discovers that his boss is using the flat for assignations with an employee, Fran Kubelik, with whom Baxter himself is besotted. It’s an acidic fable but a musical demands uplift, which Bacharach and David liberally supply. The most grating moment comes when four adulterous department heads sing Where Can You Take a Girl?, sounding less like a satire on misogyny than a surreptitious endorsement of it.

Even the form of the musical seems at odds with the subject. The show has what Mark Steyn called “the scene-song-scene-song format”, which I happen to like but which is also somewhat inflexible. After Fran has tried to kill herself with an overdose, Baxter and the doctor from the next door apartment try to cheer her up by singing A Young Pretty Girl Like You and cavorting round the apartment doing jokey business with surgical implements: it seems the wrong response to the situation. It doesn’t help that this production overloads the evening, as in the 2010 Broadway revival, by interpolating two hit songs, I Say a Little Prayer and A House Is Not a Home, from the Bacharach/David back catalogue.

Fortunately, a classic Bacharach and David number, I’ll Never Fall in Love Again, springs naturally from the action, and the Simon book is consistently witty. Simon has Baxter narrate the story, which allows him to be the first critic of his own spineless actions. There are also some vintage Simon lines, as when Baxter, after watching a desperate Fran frenziedly downing a drink, remarks: “I’d hate to see her eat.” Often thought of as a mechanical gag-dispenser, Simon has the dramatist’s ability to place a line.

But, if the show is a mixed salad, so too is Bronagh Lagan’s production. The band is too loud for a small space and the set, with a vast door that never properly closes, is clumsy. Yet Gabriel Vick brings a good deal of charm to the self-deprecating Baxter, Daisy Maywood is a soulful, well-sung Fran and there is striking support from Alex Young (“Do you like this coat? It’s owl”) as a bar-room pickup and from John Guerrasio as the disgruntled doctor. The evening has its pleasures but it seems marinated in the 60s where they did things differently.

•At Southwark Playhouse, London, until 18 February. Box office: 020-7407 0234.