So I've recently taken up an interest in Samurai Champloo (if you like anime I highly suggest watching it) but I can't seem to find many people talking about type. Anyways, here's what I think.
Fuu: ESFJ. She's caring, tries to be as law abiding as possible, emotional but not quite NF emotional. Here's a really good reference for her type MBTI: Fuu- ESFJ | Zombies Ruin Everything
Mugen: ESTP most likely. He acts quickly, doesn't think things through very well, risk taker, sharp reflexes, etc. He had a lot of confidence in his physical ability, and constantly challenges himself, usually at the expense of others.
Jin: ISTJ or ISTP ?? I have trouble with this for a few reasons. He's quiet, and I think he uses Ti, but he also seems to be somewhat traditional in the way he does things. I'd probably lean towards more ISTP honestly, but I wouldn't dismiss the idea he could be an SJ. I know a lot of people like to think he's an INTJ, but I just don't think he uses Ni often, if it all. However I'm only halfway through the series, so maybe I could be wrong.