An opportunity to whizz around on tiny trains under the streets of London has helped propel a small postal heritage charity into being a contender for the world’s biggest prize for museums.

The Postal Museum, in London, which opened last summer, is one of five museums shortlisted for the £100,000 Art Fund museum of the year prize for 2018, it was announced on Tuesday. It will compete against Tate St Ives, Glasgow Women’s Library, Hull’s Ferens Art Gallery, and Brooklands motor and aviation museum in Weybridge.

Simon Opie, director of the Postal Museum, said it was an honour. “It is quite surreal in a way … to be so new and fresh and to be shortlisted is a very welcome and unexpected surprise.”

Visitors are unquestionably drawn to the museum by the opportunity to ride on the Mail Rail, the mothballed underground network that carried post to sorting offices between Paddington and Liverpool Street stations.

“Opening up Mail Rail to the public is the thing that has really captured the imagination of all of our visitors,” said Opie. “It was the opening of something most people did not know existed. Fortunately the people who come really enjoy that, but they also enjoy everything else we have to offer.”

The museum has evolved from being a Post Office archive with 3,000 visitors annually to being a museum expecting to have first-year visitor numbers of 185,000 in 2018. It offers pillar boxes, stamps, uniforms, vehicles and more. “Through postal history you can basically tell the social history of Britain from the 16th century onwards,” said Opie.

The other four museums have been shortlisted after refurbishments. The Ferens, a star of Hull’s city of culture year in 2017, had a £5.2m refurbishment and complete rehang before it played host to the Turner prize.

The Postal Museum, in Clerkenwell, London. Photograph: Marc Aktins/Postal Museum

Stephen Deuchar, director of the Art Fund, said all five shortlisted museums were tapping into current concerns with the Postal Museum addressing our “first social network”.

He added: “The progress of Glasgow Women’s Library exemplifies the quickening march towards equality, Brooklands is inspiring the next generation of engineers, and the Ferens in Hull and Tate St Ives are galvanising their communities around visual culture. Each one expands the very idea of what a museum can be.”

The prize is given annually to a museum which has, in the view of judges, shown “exceptional imagination, innovation and achievement in the preceding year”.

The shortlist was chosen by a panel made up of the BBC’s arts correspondent Rebecca Jones, the director of the Science Museum Group, Ian Blatchford, Melanie Manchot, an artist, and Monisha Shah, an independent media consultant.

Winning is a huge deal, not just for the prestige but for the top prize of £100,000. The other four museums will each receive £10,000.

Previously known as the Gulbenkian prize and then the Art Fund prize, former winners of the award range from the enormous – the British Museum in 2011 – to the tiny – the William Morris Gallery in Walthamstow, east London, 2013. Last year’s winner was the Hepworth Wakefield.

This year’s prize will be presented at the V&A, London, on 5 July.