All I can add at present is Tom Munnelly's notes from The Bonny Green Tree:

"The motif of the suitor who, on gaining admittance to the side of his recently deceased true-love, there ends his own life is not uncommon in broadsheet balladry for such a tragic theme would undoubtedly have great popular appeal.  An interesting occurence of this motif appears in  Lord Abore and Mary Flynn  a version of  Prince Robert  which like  The Well Below the Valley  is a Child ballad (No. 87) extant only, if tenaciously, in the Irish singing tradition.

Once There lived a Captain is also something of a rarity for the only other traditional performance of the song which I have ever heard is that of Seán Ó Conaire of Rosmue, Co. Galway.  Kevin Conniff, a fine Dublin revivalist singer, now with the Chieftains, has long had it in his repertoire, his source being Seán Ó Conaire."

He also adds, re. verse 8, line 4:  "Seán Ó Conaire sings Lying dead in that room.  He also has the final two stanzas in reverse order to John's singing which makes somewhat better narrative sense."

Are you in a position to transcribe the tune?