The model that we see in the film is actually an early prototype of Mitsubishi A5M, a predecessor of A6M Zero. The unique feature of this airplane is the inverted gull wing design. The actual final designs of both A5M and A6M did not feature such wings. Interestingly, the American Chance Vought F4U Corsair did have a similar wing design, and its initial version V-166 (XF4U) was completed in 1938 by Rex Buren Beisel.
Would you like to fly the Zero?
It is not possible now, but you certainly can do it in a virtual environment! There are only a few very good WWII flight simulators:
Zero versions are only available in the War Thunder and the World of Warplanes so far. War Thunder is better than World of Warplanes in most aspects, especially physics.
You can unlock arguably the best version of Zero, A6M5 Ko by buying the Japanese Pacific Campaign (1941-1942) or earning the earlier versions of the airplane for free by playing the game.
Disclaimer: I am not affiliated with the developers of War Thunder and didn’t receive any incentive or payment from them. I am just providing you the best options to my knowledge on flying the aircraft.