When I took Russian in school I heard a story that Russian soldiers during WW2 (WW1?) would go into Paris restaurants and yell быстро, быстро, быстро!!! Meaning they wanted their food quickly, but the French took the word and named their restaurants that.
The online etymology dictionary:
http://www.etymonline.com/index.php?allowed_in_frame=0&search=bistro
says that the origin of the word is unknown.
It does recount your story that it is "commonly said" that the word came from Russian soldiers occupying Paris (in 1815, after the defeat of Napoleon, not in WWI or WWII) using the Russian word bee-stra, or "quickly", but counts this explanation of origin as "unlikely". It offers another possible derivation from bistraud, or "little shepherd" from the Poitou dialect, however, comes down as asserting that the origin is "unknown".
You might try posting this to /r/etymology if you don't get an answer here.