No, this seems like an adaption-in-spirit of Old Man Logan. It's going to have aging, retired Logan on one last mission in a very western setting.

First of all, this isn't based on Old Man Logan. Since Disney/Marvel owns the rights to most of the characters in the story. Marvel can probably make a more comic-accurate Old Man Logan movie than Fox even without Wolverine. Besides, most people who have read comics have said that this is most based on the Death of Wolverine, but disguised as Old Man Logan. Second, the director clearly said that this takes place after X-Men: Days of Future Past. So it isn't an alternative universe. Making things even more confusing.