It looks interesting but I am less sure on canonicity. The whole Four Years War seems to come from an old FASA roleplaying game as opposed to the "official" canon.
That is unclear. But I did find this reference to the Battle of Axanar:
http://en.memory-alpha.wikia.com/wiki/Battle_of_Axanar
According to that, this might have been a four year war including the battle. The war with the Klingons is consistent, but then the Wiki says that the Federation was in a "cold war" with them at that time. I give them a slide on it, because it was just before Kirk and that is when the Constitution Class starships came online. Kirk commanded one (the NCC-1701 Enterprise). Trek resources have added the Ares class ship to the fleet catalog.
I am not sure if you caught it, but in the short movie, they mentioned how the Andorians gave the Federation their phaser technology. Prior to that (Archer's ship and other Federation starships) used "phase cannons". They are shown in this clip too. The Constitution Class did not have phase cannons.
See, none of this stuff is relevant in NuTrek. We do not even know what class the new Enterprise is. We saw the USS Kelvin, and the new ship in STID, but there is no rhyme or reason to them.