There may be some things about Satoshi Kon's last words that may be puzzling to non-Japanese readers, so I'm going to attempt to clarify some of them. Note that this is not based on any kind of personal knowledge of Mr. Kon or his family, but just on general principles that are atari mae, commonly held mores and principles, in Japanese culture.
He repeats that he's ashamed of his "irresponsibility"
The word he uses in Japanese is 不義理 (fu-giri) that I have translated to "irresponsibility", which is the negative of 義理 (giri), a word that doesn't translate directly to English. It means obligation and responsibility and obedience. A Japanese person typically has giri in many ways - to obey ones elders, especially ones parents, to obey the rules of society. In the feudal era, the giri of samurai warriors first and foremost - their reason for living - was to obey and follow their lord to the death. (So an unemployed samurai or ro-nin was practically without life, soulless.) When Mr. Kon says he is ashamed of his fu-giri), he's feeling guilty of not having fulfilled is obligations, as he sees them, to his relatives, friends, co-workers, and society in general - for what he perceives as his lack of responsibility in not telling them of his illness.
Dying before ones parents is so disrespectful
This is all about the Japanese concept of 親孝行 (oya koukou), or respecting ones parents and making them happy. This concept is explained in detail on my language blog.
"I'm leaving before you"
As I mentioned in the translation notes of the document, the last sentence in Japanese is:
じゃ、お先に。(ja, osaki ni.)
If you've studied Japanese a bit or read manga, you might be familiar with another greeting said when someone is departing, またね (mata-ne). While mata-ne means "see you again/later", osaki ni means "Excuse me for leaving before you". I'm sure that he chose this term on purpose, to indicate that he is departing for another world before us, the readers.
A short history of cancer pronouncements in Japan
Mr. Kon and his wife are given the news together about his cancer. It might be interesting to know that up until quite recently until around 20 years ago) it was common for doctors to not tell their cancer patients about their illness. They would tell their relatives, but it was believed that just telling the patient that he had cancer would so devastate him that it would depress him to the point of hastening his death. (Of course, at the advanced state of his cancer, it's possible they would have told him anyway.)
In addition, I am not sure if the fact that he didn't get around to telling his parents about his cancer until it was almost too late strikes anyone as being strange, but to me, it quite fits the natural tendency or instinct that exists in the Japanese psyche about wanting to protect loved ones from really terrible news about themselves, to avoid bringing them pain.
He never says "I love you" to his wife
In the section where he thanks his wife and says she is awesome (the original Japanese phrase he uses here is 私の妻はすごいぞ (watashi no tsuma wa sugoi zo)), he never once says to her "I love you" and such. Japanese people really don't say that to each other, especially if they've been married or together for some time. (Note that Mr. Kon was 46 when he passed away; people of his generation and older generally never went for those Western type terms of affection, kissing in public and such. Younger Japanese people do sometimes, to an extent, but it's still not quite 'natural'.) But I think that the words he does use to describe his wife, and the way he talks about her, shows how deeply he did love her, and she him.
The Japanese concept of death and the afterlife
I have no idea what Mr. Kon's religious beliefs were, but I am guessing that he was a typical Buddhist-Shintoist, as the vast majority of Japanese people are. If he had been a Christian for instance, it would have been evident in his words. (My father's side of the family is Christian (still rather unusual for Japan) so I do know how Japanese Christian people speak.) So, his concept of the afterlife would have been a mixture of Buddhism and Shinto. He talks often about "going to the other side", and that's how he saw his death - as passing to another realm, a very long trip, rather than any "am I going to heaven or hell" type thoughts.(In Buddhism it's also believed that the soul hangs around for at least 49 days after death before it it becomes one with Buddha and finally goes to the other realm. In Shinto, the common belief is that the spirits of loved ones (and those of ones ancestors) are always vaguely present, and it's important to honor them.)
On a personal note, I was compelled to translate the document for a couple of reasons. First, because he was such a talented person, who died tragically young. Second, because my favorite uncle passed away earlier this year in Japan of the same cause, pancreatic cancer. His too was a swift and painful illness, though my uncle never once complained. And finally, because it is such a beautiful document.