Here’s Hugh’s probate abstract:
The Will of Hugh Hart late of Bluefield North East Division County of the Town of Carrickfergus Farmer who died 25 October 1893 at same place was proved at Belfast by William Mann of Marshaltown Carrickfergus Farmer and Thomas M'Allister of Cloughland Carrickfergus Farmer the Executors.
The will itself is on-line on the PRONI will site, and mentions his son William (who was to inherit the farm), daughter Maggie and daughter Alexandrina Jack and her son Hugh Jack.
Here’s the family in 1901:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Carrickfergus_Rural/North_East_Division/991852/
1911:
http://www.census.nationalarchives.ie/pages/1911/Antrim/Eden/North_East_Division/193195/
William died in 1916. Here’s his probate abstract. The full will is not on-line but should be in PRONI in paper format.
Probate of the Will of William Hart late of Bluefield Carrickfergus County Antrim Farmer who died 22 June 1916 granted at Belfast to Thomas G. Robinson Flesher and Hugh Hart Jack Farmer
You say that Hugh Alexander’s wife Jane died in 1903. However his death certificate shows him to be a widower. So she pre-deceased him. 2 possible deaths registered in Larne. One in 1865 aged 52, and the other in 1880 aged 81. Neither of those deaths is free to view on-line so you would need to pay to view them on the GRONI site to see if either is the correct one.
Can’t see Hugh & Jane’s marriage in the civil indexes so they probably married pre 1845 when registration began. You would need to search church records in PRONI to find them.
Hugh’s farm was plot 143 in North East Division. It was 37 acre farm. He is listed there in Griffiths in 1861. A housing estate has been built on the land today, and the modern Tudor Rd, Carrickfergus runs through the farm.