Delighted you found at least some answers on here. Scraping the barrel a bit here, but this is all I can think of.
Prison officers were sometimes recruited from ex army sources, so it is worth looking to see if Robert himself was ever in the army. Another possibility is that he was in the Royal Irish Constabulary, records are in the Dublin City library R I C Records 1816 - 1922
General registers of service, returns of personnel and intelligence notices, 1816 - 1922.
Service Nos. 1 - 4000, v. 1-2 (1816 - 1840) Reel Number: 856057
Also you could consider trying to track staff records for Richmond, Newgate etc. to see if he was working in another prison before Mountjoy.
Looking around for a birth for Robert has not turned up anything that leads directly to him, but there are certainly a lot of men of the right age to be his father of the surname Eccles in Ireland and abroad at the time.
Have you considered that he might have been an illegitimate son of either a male or female Eccles - I have a similar case with a Brabazon on my tree c. 1770 - not a trace of him anywhere, but he married one of my female line and lived right beside the main lot.
It might be worth having a look at any surviving wills or will abstracts of the Dublin, Tyrone and Clones Eccles to see if there is any mention of a natural son who might fit the bill, in the National Archives.
Mountjoy: the story of a prison by Tim Carey Collins Press, 2000, might be of interest, and it might be worth contacting the author.
PRONI holdings for this family include:
ECCLES Estate Fintona / Tyr Title deeds, etc. 1773 - 1823 D 674
ECCLES Anna Dublin Will c1816 D 2081
ECCLES family Cos. Fer,Cav,Tyr Gen. notes c1700 - 1914 T 1299
ECCLES family Dublin Gen. notes 1296 - c1963 D 2081
ECCLES family Enniskillen / Fer Pedigree 17c - 19c T 1185
ECCLES family Fintona / Tyr Leases, deeds etc 1752 - 1879 D 3850
ECCLES family Fintona / Tyr Corresp. etc 1869 - 1930 D 1385
ECCLES family Estate Fintona / Tyr Title deeds c1790 - 1830 D 1020
ECCLES family Estate Fintona / Tyr Title deeds 1659 - 1867 D 1048
ECCLES William Belfast Copy Will c1835 Mic/61add
Not digitalised, but if you did find a link to your Robert, than might make an interesting day out in Belfast.
House of Commons papers, Volume 51, 53 and 54 By Great Britain. Parliament. House of Commons (free on google books) have lots of interesting general stuff about the Joy in the period Robert was there, for background info.
The Mountjoy prison archives themselves have survived and are in the National Archive. Whether Robert turns up there is anyone's guess, but you might find them of interest. National Archives call nos.: 1/11/1-4, 9-17, 29-33, 36 for 1845-1899.
There is one curiosity in the NLI:
Photostat' copies of a family record drawn up, c. 1850, by Anna Maria Dickson, together with pedigrees of the Eccles family, compiled, c. 1800. Originals in the possession of Rev. J. M. McWilliam, Dumfriesshire, Scotland.
She is on the tree you mention here on Post 10, I think, and this could possibly be the source for that tree.
Dublin: National Library of Ireland, Ms. 8371. But it would be a seriously long shot to find your Robert there.
Sorry there is not much concrete there!
PM
