Okay, I'll take a swing at this. I am a professional ESL teacher (and school owner) in Japan.

My name is ~.
- Very easy phrase, everyone learns this in school.

Today very lucky.
- Sentence is missing a subject and a verb (I was), common of Japanese English.

Congraulations meeeeee...
- Everyone knows the word "Congratulations" and "me" and he just strings these together...

English, a little. No problem.
- This is a direct translation of the Japanese (英語は少し). Again, we see the same mistakes - no subject, no verb. The word for "a little" is generally used as an adverb in Japanese, so he's placed it after English because that's where it would fall in Japanese. Then the set phrase that everyone has picked up - "No problem."

Only so... - I actually have no idea where he picked this up and it makes me think that he MIGHT be faking it. My students would never say this, but he might have picked it up as an alternative to "Only that" (それだけ in Japanese).

I have a pen.
- This is a very, very, very common phrase taught in Japanese schools when they're first working on English grammar. He obviously had picked up this little pen/apple joke somewhere though.

I have apple.
- He forgets to say "an apple" here... this is because in the previous sentence, he used a sentence that is drilled into their heads. He can say "I have a pen" but doesn't truly understand why we have to use the article "a" there. So when he says apple, he is winging it and misses the article (because articles don't exist in Japanese).

A...E... apple pen.
- I have no fucking clue what he is trying to say in the beginning there. Just some exclamatory sounds, I think. Like "Wahoooo apple pen." Then he makes an obscene gesture...haha. I actually don't think he knows it's obscene, though.

sonna kanji (そんな感じ？)
- After this, he flips back to Japanese because he's nervous. He says, "Like that?" (literally) but it means more like "Is that okay?" or "Is that the right answer?" (in a joking tone)

un...unnnn....un... no problem.
- These "un" sounds are very Japanese and they basically just mean "uh uh...hmmmm... uh huh... " The fact that he flips over to Japanese immediately shows he doesn't really understand English, but he also could catch enough of the interviewer's question to guess what he was saying.