The German wikipedia page of the Archduke gives an even higher number. Specifically it says that he shot on a single day in June 1908 2763 black-headed gulls. They even cite a book as a source just for this claim, so it seems to be a reliable number. However, I wasn't able to find anything on how or where he accomplished that number.
However, what John Dallman said is true. He usually didn't reload the gun himself. In this picture, which is titled Franz Ferdinand hunting in 1912, you can see that in addition to him there are at least to other people holding a gun (maybe even the guy on the right). Although the description of the picture doesn't specify that explicitly, those guys are most likely his servants responsible for reloading his guns while they hand him a loaded one. That way he was able to shoot with a high frequency. (This was in general how the higher aristocracy at the time was hunting.)
While by hunting birds you can get a large body count in a short amount of time, Franz Ferdinand shot also many larger animals like deer. The general strategy of the aristocracy at the time was to designate certain areas as your hunting grounds and to appoint several people who's only job it was to make sure that when you go hunting there is enough game. Usually this resulting in an substantial over-population of game in these hunting grounds. Then on hunting day, the servants looking after the game sent you to a promising spot, where game was very likely to pass (That's what actually what you can see in the picture). When the hunt began some people went with dogs into the woods and made some noise, which made the game moving. That way you had the chance to kill several animals during a day. (The Archduke didn't bring 3-4 guns with him for nothing.)
Actually, the trophies of his hunting trips survived until today; they are still in his former hunting castle. For instance, just watch this video of somebody visiting it: Trophies of Franz Ferdinand (the relevant part starts at around 1:30).
Finally, while Franz Ferdinand was excessive in his hunting, it is just the high number of kills that made him stand out from his fellow aristocrats. For instance, according to German wikipedia, emperor Wilhelm II. shot around 46000 animals during his live. Apparently, aristocratic duties were no obstacle to going hunting a lot.
Btw.: machine guns and armoured vehicles are not really effective for hunting. In contrast to humans in a war, animals tend to run away when they hear shots, and with a machine gun only the first shot tends to be precise. There is this surreal episode where the Australian army tried to decimate the Emu population. It is called Emu war and was a complete failure. With respect to hunting them with a vehicle, the corresponding wikipedia page explicitly states that 'At one stage Meredith even went so far as to mount one of the guns on a truck: a move that proved to be ineffective, as the truck was unable to gain on the birds, and the ride was so rough that the gunner was unable to fire any shots.'