Since the place and cause of death are not specified, there are a number of places you could look.
Frank and Andrea Cook's Casualty Roll for the Crimea lists battle casualties which appeared in the London Gazette in 1854 and 1855 . . . book available at a number of used book sellers.
If the person actually made it to the Crimea, he might be listed on Yesteryear's Genealogy's Medal Roll for the Crimea CD, which has PDF photocopies of I believe about 7000 pages of medal rolls separated by unit . . . sometimes first names are listed, sometimes not.
Join the Crimean War Discussion Group at Yahoo Groups and post your query. There might be someone with a passion for the Royal Artillery that might have the information.
Best, but most difficult . . . go to the National Archives at Kew (or pay a professional researcher) to check the muster rolls there . . .