I think your idea to make it about misunderstandings similar sounding Japanese words is great. Even if you aren't used to making puns, you must have encountered a lot of confusingly similar words while studying. The audience can probably relate to that too.
Besides making a skit about making a skit, here are some other ideas on how you could include Japanese in your skit:
Two people are taking a Japanese test. The funny man is clueless and keeps whispering to the straight man asking for answers, but he keeps misinterpreting the straight man's answers. Or the straight man asks where they could get more paper for the essay question, and the funny man plucks a strand of hair out of the straight man's head. It's the same as writing a skit about a skit, but it might be easier because you don't need a lot of connection between the puns. It could also not be a Japanese test, but a test on another subject like History, only that they are taking the test in Japanese.
The straight man is taking the funny man over to a Japanese person's house. The straight man give the funny man advice on how to introduce himself, but he keeps botching it up. "ぼくはマイクともうします" becomes "ぼくはマイクとおもいます" (I hear it is a common mistake, especially when you learn it through the ear) or "ぼくはジョンのあにです" becomes "ぼくはジョンのおにです". They are brought to a Tatami room.
Straight man: "This is called a Tatami"
Funny man: "Tataki"
Straight man: "No, tatami"
Funny man: "Tatari"
Straight man: "No, TA-TA-MI, can you say that?"
Funny man:
"To-to-ro"
Straight man: "You're getting even farther! Never mind"
Straight man: "In this kind of room, we should sit in Seiza-style."
(Straight man sits with legs folded up.
Funny man starts sitting in various strange poses, reminiscent of Greek heroes and monsters.)
Straight man: "What are you doing?"
Funny man: これは"Orion" の星座 (せいざ、constellation)。 これは"Pegasus"の星座。
Straight man: ちがうよ。正座は、足をまげてすわることだよ。
Funny man: ああ、そうか。(sits with crossed legs, indian style)
(Finally, they are sitting with folded legs. After a while, the straight man points at the funny man's legs and asks)
"いたくないですか?"
Funny man: "はい。いたくないです。"
Straight man: "本当?"
Funny man: "(in a whisper) いたくないです。この部屋にいたくないです。出ていっていいですか?"
Straight man: "ばか!"
Funny man: "Oh! I guess that's rude. いたいです。すごくいたいです。いたーいなー!"
(A moment later they are out of the room)
Funny man: "why did they take us out of the room? I said I wanted to be there"