I was wondering how old Logan or James Howlett was in The Wolverine, not at the flashback but at the prime of the movie.
About 181 years old
In X-Men Origins: Wolverine (set before The Wolverine), we see a young James Howlett in the year 1845 when his mutant gene is activated for the first time.
Let's assume that the fact that he had a fever meant that his healing factor (which is what keeps him young) hadn't kicked in yet, so he's as old as he appears. Actor Troye Sivan was 13 at the time of the film's release, so if the character he plays is the same age, then his birthdate is in 1832.
The Wolverine was released in 2013. If we assume that it takes place in the same year, then that would mean that he is about 181 years old.