Is third a common noun

Yes, the noun 'third' is a common noun, a Word for any third of anything.

A proper noun is the name of a specific person, place, thing, or a title; for example:

• William the Third, King of England and Ireland 1689-1702
• Third Street, New Orleans, LA or Third Avenue, New York, NY
• Third House Inn, Vancouver, BC Canada
• "Shrek the Third", 2007 animated DreamWorks feature movie