You need more information in order to search.John Armstrong is a relatively common name and with 111,322 officers and men sent to the Crimea, I am sure there will be many John Armstrongs. If you can find out what Regiment he served in it would be a great help . . .
From my records of about 3300 (of the 111,322) men:
There was only one John Armstrong who served in the Light Brigade during the war years. He served in the 8th Hussars with Regimental number 1048. He enlisted late in 1847 which would indicate he was probably born c. 1827 give or take 5 years. He served in the Crimea, spending some time in 1855 as a Hospital Orderly. He was not present at Alma, Balaclava or Inkerman, but was awarded a Crimean War Medal with a Sebastopol Clasp.
Don't know if this was your relative, but if not, you can eliminate the 4th Light Dragoons, the 8th Hussars, the 11th Hussars, the 13th Light Dragoons, and the 17th Lancers from the list of regiments serving in the war which he might have been in.