The Wearing of the Green Song Book, dated 1869 (mentioned by Goose Gander above), can now be seen at Google Books, and it contains the lyrics to OLD SKIBBEREEN. [Click for title page.] I haven't compared word for word, but I assume the lyrics are the same as those posted by Goose Gander above (although we aren't certain those lyrics came from the same edition; we know that several editions were published).

This pushes back the origin of the song a bit.

Complete citation:

The Wearing of the Green Song Book (Boston: Patrick Donahoe, 1869), page 208. [No author or editor of the book is named as such; presumably the publisher is also the editor. The song is credited to Patrick Carpenter.]