I strongly suggest you run a name check for all James Fox’s being awarded the Crimea Medal and clasps in the UK, Military Campaign Medal and Award Rolls, 1793-1949 database on Ancestry. It will then be possible to establish exactly how many possible soldiers you could be looking at and the regiments they served with.
The difficulty you will face is that prior to 1883 soldiers’ papers, such as they are, were only preserved for soldiers discharged to pension. If they were discharged for any other reason, purchased their discharge, dismissed etc., no record will exist and you would have to resort to searching whatever regimental documents, like Muster Books & Pay Lists or Description Books, to piece his service together; none of those records are online.
There are only a handful of James Fox soldiers discharged to pension in the 1854-1872 time frame, including this one who was discharged to pension after falling ill in New Zealand.
James Fox, Regimental No 3856
Born Parish Englishtown, Town of Englishtown, Co Limerick
Attested for 50th Regiment of Foot 27 August 1853 age 17yrs 3mths (calculated year of birth 1836)
Received Crimean Medal + 3 clasps and Turkish War Medal.
The National Archives reference: WO97 / 1544 / 189