The new trailer is out.
I can think of a couple ways that this movie might conflict with canon. First, this is a movie about Kirk's first mission on the Enterprise, and it features other familiar faces like Spock, Scotty, and Chekov. But on the TV show, Kirk was the captain while Chekov was still an ensign. This would suggest that Chekov is at least 10 years younger than Kirk. So would Chekov be still in elementry school during Kirk's first mission? Other possible explanations: Chekov started as an enlisted man, got caught up in a time loop, went into a coma, or the guy in charge of promotions at Starfleet HQ is biased against Slavs.
Still, you shouldn't let inconsistencies ruin this movie for you. "Patton" was a good movie even though George C. Scott didn't look or sound anything like the actual general.