I remember first reading the MMM site thinking "ok this doesn't apply to me here in Japan.... yes, this also doesn't apply to me..." etc.
The obvious ones of course are the "no cars due to awesome public transport" and lack of clothes driers/dishwashers/etc. One more I have found is that you just have less time to spend money here - after a 10-14 hour day, drinks with workmates, weekend work, etc often it is an effort to think of ways to spend more money.
On a cultural note, Japan is divided with regards to its attitude towards frugality ("setsuyaku"). On the one hand it has a bad reputation, a strong association with "cheap" and "poor" (not to mention that it is the kind of thing one is forced to do during a war) and many people don't want to have anything to do with it. On the other hand, there have been reality TV shows about frugality here, people meticulously fill out their "kakeibo" (budgeting notebook, often as an app these days. Think YNAB), etc so there exists at least a bit of a movement here.
There are also a few cultural reasons Japanese do not make good Mustachians. The first one that comes to mind is how, when leaving a restaurant, it is custom to leave some food and drink behind unconsumed. If a dish of food gets eaten down to the last then another one must be ordered, according to custom. The signal to leave the restaurant is that everyone has half a glass of beer remaining and there are about five dishes on the table each with one or two pieces of food remaining on them lol.
While we're doing a comparison, at least in the West people have a clue about investing. In Japan, it's either "JGBs 100%," or "JGBs 90%, AUD carry trade 10%" lol. Maybe that will change with Abenomics/low bond yields/higher stock prices due to increased money supply/NISA/etc but so far the takeup has been slow.
The hardest thing about being a Mustachian in Japan (Tokyo) would have to be the choice of restaurants and delicious foods.