Meaning of art in this context

Yes, it's a reference to the Latin (originally Greek) aphorism "Ars longa, vita brevis" commonly translated into English as "Art is long, life is short."

"Art" here doesn't mean the fine arts but technique or craft.

Technique/ craft can mean different things to different people, and thus one's interpretation will also vary.

But the poem "The Last of the Light Brigade" talks about the Crimean War, in particular the soldiers who stood to greet its horrors. As you can imagine, they were afraid for their lives and concerned about their mortality.

The word"art" here, which pertains to something more lasting, serves as a contrast to short human "life" that the soldiers quite naturally valued.

Your question is answered by the very next line of the poem:

They felt that life was fleeting; they knew not that art was long,
That though they were dying of famine, they lived in deathless song.

Kipling here is referring to Tennyson's famous poem The Charge of the Light Brigade. The art is Tennyson's poem. You can see from the other answers that this isn't exactly what ars longa, vita brevis originally meant, but Kipling is playing with language a little here.

Ars longa, 
vita brevis, 
occasio praeceps, 
experimentum periculosum, 
iudicium difficile. 

art is long,
life is short, 
opportunity fleeting, 
experiment dangerous, 
judgment difficult.

It should be noted that ars does not signify fine art but the art of some skilled activity – in this context, the art of medicine.

For further information, see this source.

It's a reference to the Latin ars longa, vita brevis - art is long, life is short.