The tune is a varient of "Divers (or Diverus) and Lazarus" (of which there are many) and so predates the words to "Star of the County Down". Maybe the question should be "What other songs use the same tune as Star of the County Down?"

No idea what the tune is called - most people will know it by the name of the song they sing to it. I have "Divers and Lazarus" itself, "The Unquiet Grave" (another Child ballad) and aleast two other songs that all use varients of the same tune.