The first building to go up on what is now Russell Court/Russell Street, in 1835, was a brewery – they had their priorities right in them days! Whatever it had been called before, by 1859 it had been named after the Battle of the Alma (September 20, 1854), which is often considered the first battle of the Crimean War (1853–1856). A new brewery was built behind a few years later, and the old brewery building became the brewery tap (which had been licensed as a beerhouse since at least 1860).[1] 
The present sign depicts a British soldier, stood beneath a Red Ensign (which is primarily a non-military naval flag, so I’ve no idea what it’s meant to be doing here), smoking a pipe and surveying the field after the battle. But is it any old soldier? Well, although not depicted in ay great detail, he does seem to be wearing the uniform of the Royal Welch Fusiliers as it was at the time of the battle. One of their number, Colour Sergeant Luke O’Connor, although himself wounded, continued to carry the Queen’s Colour throughout the battle. For this, and for other actions during the war, he was later awarded the Victoria Cross, the first soldier to receive this honour. So perhaps this is him, the red ensign is standing in for the Queen’s Colour, being rather easier to paint.[2] Sgt O’Connor has no connection with Cambridge, though. But the commander of the 1st Division (Guards and Highland brigades) at the battle was Prince George, the then 2nd Duke of Cambridge (fourth creation, 1801). If there is an original painting on which sign is based, and if I ever manage to track it down, that may clear up a few things.
References
[1] Flood, B., Cambridge Breweries, Cambridge Society for Industrial Archaeology and CAMRA, 1987.
[2] See this photograph of a modern reenactment group, including the Queen’s Colour.
