"won runner-up" is correct and usable in written English.
You can use it when referring to someone who did not win first prize, but still achieved a high ranking. For example, "The student was awarded second place and won runner-up in the spelling bee."
After some tough deliberation, ISI Technologies won runner-up, and the 2014 TechCrunch Disrupt NY Startup Battlefield winner was Vurb.
We saw it simmering at Disrupt this year; Lit Motors won runner-up, and I found Hardware Alley more interesting than almost all the software startups.
The move to veto the first of these apps will now set the tone for those who would have followed – including, perhaps, the forthcoming app from a team at this year's TechCrunch Disrupt SF hackathon, who created a prototype app with similar functionality they called Blitz, which won runner-up.
His app Band of the Day usurped the traditional music blog, running for five years and winning runner-up to Instagram for Apple's App of the Year in 2011.
His pictures were nominated for the Pulitzer Prize, and his single image won a runner-up award.
"Adaptation," starring Nicolas Cage in a double role and directed by Spike Jonze, won the runner-up Silver Bear award.
Livingstone shows me the "deluxe Monopoly set" he won as runner-up in the 1975 British Monopoly championships.
When we posed the challenge last year and Jim Gaffigan won, the runner-up was Michael Che, a performer with a year's experience onstage.
Michael Gaskell won the runner-up prize of £8,000 for his portrait entitled Harry, and the third prize of £6,000 went to David Eichenberg for Tim II.
Heller's first clip was an interview with a "slightly mad teenager" who lived next door, for which she won a runner-up prize for feature writing from British Vogue.
A major hit with critics at Cannes, where it won the runner-up prize to the Palme d'Or, Son of Saul is one of the most intense Holocaust movies ever attempted.
