In the movie, his power does appear to prevent aging.
In fact, it seems to be a pretty central theme to the movie. Yashida explicitly states that what he is offering Logan is an end to his "eternal" existence"
"I can end your eternity. Make you mortal."
He's quite clear that he's offering to allow Logan to grow old, and die of a "normal" death.
Wolverine has not aged at all since he first met Yashida in World War II. It can be assumed from the ending that this "immortality" is achieved by his mutant healing factor "healing" the aging.
As noted in the comments above, this is not necessarily consistent with the comic book universe(s), but the repeated discussion of Wolverine's immortality, and the possibility that he will (or has) run out of things to live for, makes it pretty clear that, as far as the movie is concerned:
Yes, it makes a kind of sense (as much as anything in the movie does) for Yashida to "heal" his old age.