Is the word run a noun?

Yes, the Word 'run' is both a verb and a noun.

The noun run can be an act of running, as in, 'I'm going for a run.', or, 'Mitt Romney is taking a run at the Presidency; a migration of fish, or fish in the process of migration; a score made in Baseball; something that flows in the course of an operation or during a particular time, 'The first run of widgets had a defect.' or 'We saw the buckets in place for the annual run of sap.'; a continuous period or series, as a run of bad luck or a long run on Broadway; a rapid passage up or down the scale by voice or instrument; a ravel in a knitted fabric such as hosiery; a painting defect caused by excessive flow.