I found this probate abstract:
The Will of William Mulligan late of Mullineany County Fermanagh Farmer who died 7 November 1895 was proved at Armagh by John Mulligan of Mullineany Farmer the sole Executor. Effects £253 4s.
The will itself is on-line and names his wife Rose Ann, so it does look to be the right family. The will also names sons John, Thomas, William & Robert as well as a daughter Mary Jane Thompson.
http://www.proni.gov.uk/index/search_the_archives/will_calendars.htm
This looks to be the family in 1901:
http://www.census.nationalarchives.ie/pages/1901/Fermanagh/Doon/Mullyrenny/1357708/
and in 1911:
http://www.census.nationalarchives.ie/pages/1911/Fermanagh/Doon/Mullyneeny/523616/
Mullyneeny is in the parish of Kinawley. The Church of Ireland records there start in 1761 so you may be able to get back a bit. There’s a copy in PRONI.
Griffiths Valuation for 1862 lists William Mulligan living on plot 1 in Mullyneeny. That was a farm, and just under 11 acres.
http://www.askaboutireland.ie/griffith-valuation/index.xml
That farm today is on the north side of the Derrylin Rd at the junction with the Stragowna Rd.
The revaluation records show a William Mulligan replacing John Mulligan on the farm in 1924.
http://www.proni.gov.uk/index/search_the_archives/val12b.htm
William seems to have died in 1956. His will is not on line but should be in PRONI. Abstract:
Mulligan William Finlay of Fairfield Mullyneeny Derrylin county Fermanagh farmer died 16 October 1956 Probate Belfast 13 December to Robert Willis Mulligan farmer. Effects £665 14s. 8d.