The lyrics posted at the beginning of this thread appear in

Timothy Daniel O'Sullivan, Dunboy, and Other Poems. (Dublin: John F. Fowler, 1861), page 130, which you can see at the Internet Archive.

Note the author’s name given there is O’Sullivan, not Sullivan—although many later publications refer to him as Sullivan.

Note that although the spelling is “hurra,” the rhyme scheme seems to require the pronunciation “hurray” or “hooray.”

Another copy of the title and lyrics, identical to those above, but with the note: “Air—We’ll never get drunk again.” appears in: Songs of Ireland and Other Lands (New York: D. & J. Sadler & Co., 1847), page 35. No author or editor is named.

Lyrics appear in several later publications.

There is sheet music at Baylor University, with the title OLD IRELAND, BOYS, HURRAH!, words by T[imothy] D[aniel] Sullivan, Esq., music by John Mac Evoy, published by Wm. A. Pond & Co., New York, ©1871. Some minor changes in the words are made, probably to more strictly fit the rhythm of the tune, and a chorus is added:

Hurra! Hurra!
We’ll toast old Ireland, dear old land,
Here’s Ireland, boys, Hurra!

I don’t know whether the tune in the sheet music matches what people actually sing today.