You didn't say when and where Richard was born but this is what I found.
A Richard Sharkey b. abt. 1834, Ireland, bootmaker, arrived in New York City 12 Jan 1854 on the ship Jacob A. Westervelt. Port of Departure=Liverpool, England.
Underneath Richard's name was a Mary Costigan, 19, spinster. Don't know if she became Richard's wife or not. Source: New York Passenger Lists 1820-1957, Ancestry.
Richard, wife and daughter in 1860 were living in New York Ward 21, District 14, New York Co., NY.
Richard Sharkey, b. Ireland, 25 yrs old, shoemaker.
Mary, wife, b. Ireland, 21.
Elizabeth, b. NY, 1.
James Connor, b. Ireland, 17, apprentice to shoemaker.
A Richard Sharkey, 26 yrs old, born abt. 1835, enlisted 25 Dec 1861 in the Union Army, New York City, NY. He enlisted in on 30 Dec 1861. Mustered out 25 Jan 1863, Fort Monroe, VA. Source: New York: Report of Adjutant-General, U.S. Civil War Soldier Records & Profiles, Ancestry.
In 1870 Richard and family were living in Little York, Nevada Co., CA.
Richard Sharkey, b. Ireland, 35, boot & shoemaker.
Mary Ann, wife, b. Ireland, 31.
Elizabeth A., b. NY, 10.
Mary, b. NY, 8
Robert, b. CA, 5.
Helena, b. CA, 3.
Margaret, b. CA, 1.
In 1880 Richard and family lived in Sierra City, Sierra Co., CA.
Richard Sharkey, b. Ireland, abt. 1835, bootmaker.
Mary Ann, wife, b. Ireland, 40.
Elizabeth A., b. New York, 20.
Mary E., b. NY, 18, "out at service".
Robert, b. CA, 14.
Elanor, b. CA, 13.
Margaret A., b. CA, 11.
Joseph F., b. CA, 9.
George, b. CA, 7.
William R., b. CA, 4.
Charles L., b. CA, 2.
Albert H., b. CA, 1 month.
Mary A. Sharkey, widow filed for Richard Sharkey's Civil War Pension in California. Source: Civl War Pension Index: General Index to Pension Files 1861-1934, Ancestry.
I didn't find anything for the John C. Fremont Post 152 except on a list of all GAR organizations in California. You can contact the Sierra County Historical Society, Sierra Heritage Historical Society or the Downieville Museum. Their contact info is at www.cagenweb.com/sierra/Organizations/Organizational.html . Successful Searching!