^
^^If the bulk of this film is set in 2260, then Chekov would only be 15 or 16 (going by the popular agreed-upon timelines). In Who Mourns for Adonais, Chekov said he was 22 years old. That episode is from the second season, which most people agree took place around 2267.
I personally think the majority of this film will take place in the months leading up to Kirk taking command -- sometime in 2264 or 2265, which would make Chekov 19 or 20 for that part of the film.
I think this is is some ways an "origin film" for crewmembers of the Enterprise, so we will perhaps see some parts of this taking place 5+ years prior to TOS. However, I have a feelingthat this will mainly be the story of how that TOS crew all came together to be on the Enterprise -- not just their separate origins, but their origins as a crew...and I think that happens much closer to the TOS timeframe.