Not only is it very difficult to date the ending of the realm of Rhudaur, but it's actually difficult to define what 'ending' might mean in this context. We do know that the line of its original rulers, the Dúnedain, quickly came to an end, and it was taken over by a tribe of Hill-men who allied themselves with Angmar. However, Rhudaur certainly continued as a political entity past this point.
It is last mentioned in annals in the year III 1356, when Argeleb I, then King of Arthedain, fell to its forces. Its ally Angmar enjoyed an ascendancy over the following centuries, so presumably Rhudaur also thrived at this time. Eventually, in III 1975, Angmar fell to the Gondorians, and it seems fair to speculate that Rhudaur's end came at this time, too.