Yes, there is a book called "Old Families of Larne & District" compiled by George Rutherford and edited by R.S.J. Clarke that has listings of the cemeteries around Larne.
In the KILWAUGHTER CEMETERY at Rory's Glen, which is the newer cemetery that is a bit down the road but within walking distance of the castle gates (not within the castle grounds), there is one Agnew listed under the a monument for William Caldwell who died 1855, whose grandson was George Arthur AGNEW.Also, on a monument for William Magee there is mentioned a Margaret AGNEW, who was a daughter of William and Jane Magee.
Did your ancestor have a married/maiden name?It states at the beginning of the list here in this book that all the stones with death dates BEFORE 1900 have been recorded, so it just could be that your ancestor's is too new to be in the book.
In the Kilwaughter Old cemetery within the castle desmesne, there are surprisingingly few Agnews. The book states that "burials ceased here in the middle of the twentieth century and the graveyard has become very wild and overgrown.All inscriptions have been copied."
KILWAUGHTER OLD CEMETERY
AGNEW [Green slate next to Elizabeth Agnew d. 1811.]Erected by W. Agnew in memory of his son William who died in infancy.
AGNEW [Large slate with arms:a chevron, in chief two gillyflowers, in base three saltires couped barwise.Crest:an eagle (?) displayed.Mottor:Concilio non impetu.]Erected by John Agnew in memory of Elizabeth Agnew who departed this life June the XIth AD MDCCCXI aged XV years.
O happy youth with graces blest
To whom the Lord has given rest.
Also his wife Margt. who departed this life May 1825 aged 66 yrs.Also John Agnew who died Mar. 1839 aged (78) years.
AGNEW [fallen forward] Erected by James Agnew of Cambridge St., Belfast, in memory of his beloved mother Margaret Agnew who departed this life on the 12th day of October 1863 aged 64 years.Also his beloved father Samuel Agnew who died on the 6th day of February 1871 aged 74 years.Also Samuel, James & Mary Jane Agnew, 3 children of James Agnew, all of whom died in infancy.
AGNEW [Slate propped against north wall.}Agnew Agnew died 2nd June 1896 aged 82 years.
AGNEW See BAILIE
BAILIE[Lead lettering on white limestone panel set into standstone, in a low railed enclosure.}Arected by Robert H. Bailie of Summerhill, Dunadry, in memory of his aunt Jane AGNEW, formerly of Richfield, Ohio, America, who died at Kilwaughter 30th April 1871 aged 77 years.Also her only child Jane who died at Richfield 2nd Jany. 1867 aged 29 years.Jenkins.
LARNE MCGAREL CEMETERY:
AGNEW[Lead lettering in white limestone panel within standstone surmounted by urn and enclosed by iron railings.]Erected in memory of Robert Agnew, Larne, who died 6th Nov. 1880 aged 52 years.Also his wife Jane Agnew who died 18tjh Oct. 1881 age3d 42 years.Here also are interred the remains of their son Joseph who died in infancy.Also their son Samuel who died 24th May 1890 aged 31 years.
[The will of Robert Agnew, late of Larne, blacksmith, who died 9 November 1880 at Richhill, county Armagh, was proved at Belfast 9 May 1881 by the oath of John Fullerton of Larne, grocer, one of the executors.Effects under 1,000 pounds].
LARNE and INVER OLD GRAVEYARD ST. CEDMAS
(Although this is a Church of Ireland, many Presbyterians are buried there).
AGNEW [Table stone of polished granite in failed enclosure.]In memoriam Patrick Agnew of Larne and Kilwaughter died 25th October 1842 age 84 years.Eliza McNEILL, his wife died 8th May 1851 aged 66 years.Anne WILSON Agnew their daughter died 23rd October 1887 aged 62 years.Jane CALWELL, their youngest daughter and the wife of John MCCAULAY of Red Hall, died 26th June 1899 aged 71 years and John Macaulay, formerly of Red Hall, died at Cheltenham 16th Setp. 1912 in his 90th year.
[Letters of administratrion, with the will annexed, of the personal estate of Anne Agnew late of Larne Harbour, county Antrim, spinster, who died 23 October 1887 at same place, were granted at Belfast 3 February 1888 to Jane Callwell Macaulay of Red Hall in said county (wife of John Macaulay), the universal legatee.Effects 2,546 pounds.5s.]
AGNEW[During work which began on the chancel in 1879 the four archangels, Michael Gabriel, Uriel, and Raphael were painted in oils on the east wall by Cox, Buckley & Co., London.On a small unobtrusive shield appeared the memorial dedication.All is now gone.All is now gone.(See Fair:To This you Belong, p. 29.)To the glory of God and in loving remembrance of Patreick Agnew of Ballasavage and Larne, died October 25, 1842.Also of Eliza his wife died May 8, 1851.And of their daughter Ann WIlson Agnew, died October 23 1887.The memorial is placed by their daughter, Margaret St. Leger Mitchell, Hariette Robertson Cunningham, Matilda Agnew, and Jane Macaulay.1889.