I definitely think he's experiencing a mid-life crisis in every way. With that being said, however, there IS more to it than simply that.
Dmitry obviously lacks the emotional connection to his wife; therefore, he searches elsewhere to find it, and he finds it in Anna. When I first read this story, I was a bit suprised that Dmitry pursued a relationship with Anna because she WAS so emotional and different than Dmitry; however, he was so drawn to that. Perhaps he wanted Anna to be able to bring out that side of Dmitry that he needs to feel.
I've often heard it said that people do not cheat in relationships UNLESS they are missing something very important in their other relationship (and it is not sexual in nature. It normally has to do with emotional closeness). I believe this to be VERY true. Dmitry was determined to make his relationship with Anna work because he had finally found someone who could help him bring out the feelings he had repressed for so long.
Dmitry is certainly at the right age for a mid-life crisis. However, many men at that stage become more promiscuous. Dmitry is, in fact, becoming less so. I wouldn't characterize his change in behavior to the traditional "mid-life crisis" conflict, but rather to a coming of age. It may have taken him half his life, but Dmitry is finally maturing. He has spent his life distanced from his own emotions and from other people. He has enjoyed casual relationships with many women, but has always kept in control, not allowing them or the relationship to affect him personally.
Anna is different, though. For the first time, Dmitry understands what it is like to be out of control and to want more than he can have. He has to learn patience and restraint. In order to remain in a relationship with Anna, he has to allow her to have power. She must make the decisions that balance their relationship with her marriage, and she must weigh all the individual risks and rewards in planning their meetings.
So, why Anna? What does she bring to him that other women have not? I think the answer is that Anna is Dmitry's foil - she has the qualities he has been lacking. She is soft, gentle, emotional, and full of a moral center that he has always lacked. While he may refer to her as "pathetic", I think this is what really draws him to her - he wants that emotion that he lacks.