What is the song called, and where did you find it?

See Song LV, 'Paddy O'Rourke, or The Pig Under the Pot' (beginning 'When I was a young man in sweet Tipperary'; no music given) in Ashburner's new vocal and poetic repository (Ulverston: George Ashburner, 1807) 126-130
http://books.google.com/books?id=IbQDAAAAQAAJ

To the tune 'When I was a young man in sweet Tipperary' is a short piece, 'At the side of the road, near the bridge of Drumcondra' (music not given), in Thomas Crofton Croker, The popular songs of Ireland (London: H Colburn, 1839) page 81.
http://www.archive.org/details/popsongsireland00crocrich

Another song called 'Paddy [or Billy] O'Rourke [or Rooke]' beginning 'I greased my brogues and cut my stick' is number 2101 in the Roud Folk Song Index. Various broadside editions can be seen at  Bodleian Library Broadside Ballads.

Two tunes called 'Billy O'Rourke's' can be seen at The Fiddler's Companion:
http://www.ibiblio.org/fiddlers/BILL_BINS.htm#BILLY_O%27ROURKE%27S_JIG_[1]
Follow the cross-references for further cross-references: As a compendium uncritically compiled, some of the information at that site is misleading or wrong, so don't take any of it for granted. Always cross-check. All the tunes given may or may not be sufficiently similar to be considered related, but presumably the one you want will be an example from Ireland, the earlier the better if 1783 is correct. The entry for 'Johnny McGill' (Burns used it for his song 'Tibbie Dunbar'; the tune may originally be Scots) may prove useful:
http://www.ibiblio.org/fiddlers/JOHNN_JOHNS.htm#JOHNNY/JOHNNIE_McGILL/MACGILL

Tunes at FC are in abc format, convenient for posting in the Forum and easily convertible to midi if eventually selected for inclusion in the DT. 'Tibbie Dunbar' is already in the DT, of course, with a midi, though no source for the latter is named.

For further titles and references, see the late Bruce Olson's website:

http://www.csufresno.edu/folklore/Olson/IRTITLE3.HTM (under 'Blarney Castle')

A wide range of tunes in abc and midi formats can be found via http://trillian.mit.edu/~jc/cgi/abc/tunefind