We're not told exactly when the Bridge was built, but it is known that it was constructed by the Men of Arnor, so the latest possible date is III 861 when the North-kingdom was divided. Much earlier than this, in the late Second Age, it is known that at least one great bridge was constructed to allow the passing of the armies of the Last Alliance. It is possible that the Bridge of Stonebows was erected for the same purpose, which would date it to II 3430 or shortly afterwards.
However we date the Bridge, it was remarkably old: at the time of the War of the Ring, it had definitely stood for more than two thousand years, and it may have been a thousand years older even than that.
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