Also plugging away guiltily...
The Persona series is also a good game if you want to learn a bit about culture/architecture/etc.
Of course, I'm going to add the disclaimer here that no matter what game you play, these are games, and what you see/hear is going to be 'filtered'; in the same way, playing the Sims or Bully can provide some cultural information about America, but it has to be taken with a grain of salt. There may be something telling about them, but they also aren't completely representative of the culture, itself. However, the fact that a game is/isn't popular can also speak for the culture, as well. Etc. etc.
That out of the way, back to Persona.
I enjoy the series for a lot of reasons, but I do think it can be educational in terms of learning a bit about Japanese architecture, culture, humor, values, etc. You may have already played it, or heard of it. It usually takes 50 - 80 hours to complete a game, but I think it's pretty well worth it. Additionally, as with every Japanese game, you do have to keep in mind that it's been translated, so humor and other things aren't going to match exactly with the original source material.
I also like Danganronpa; in spite of it's flaws, it has some really great philosophy to it. It doesn't necessarily tell you much directly about Japanese culture, etc., but themes like hope, despair, etc. seem to be a really popular theme in many Japanese games nowadays, and I think the message it has for players is particularly powerful in some cases.
I.e. - You can learn about Japanese culture and such through games, but what is perhaps more useful is understanding how the culture presents itself in games, how it understands foreign cultures, and how these games speak in particular for gaming culture. I don't know if any games in particular that focus exclusively on culture, but i think as you play Japanese games you may notice certain things that reoccur in the gameplay design, settings, etc.
Sorry, I'm pretty damn scatterbrained right now, so this isn't the quality of post I'd like to write. >.<; If you haven't visited Japan, but would like to, definitely do so! I took a vacation not long ago, and it's very nice.
I think if you play any game, you'll see a lot of the same things, in terms of character designs (personality), plot themes, etc. Although it may not say a lot about Japanese culture directly, they do speak indirectly for themselves.