So, this is a total anecdote, but I recently moved to Tokyo from SF, and while I don't think it's necessarily more expensive to live here, I wouldn't say it's a bargain by any measure.
My rent is comparable to what I paid in USD in various areas of the SF peninsula. However, move-in costs were outrageous. I have a pet, so that increases it some, but everything here for rent is a multiplier, ie.:
- I had to put 3x monthly rent down up front for security deposits (due to pet, average is 2 months)
- Pay the first two months rent up front
- Pay the real estate agency that helps you find an apartment (pretty much required) one month rent
- Pay a "guarantor" company half a month's rent for some kind of renter's insurance I guess
I'm lucky that I didn't have to pay "key money" for this apartment, which is effectively a bribe to the owner that you'll never see back. I also won't ever see my real estate fee or guarantor money again, and it's common to lose at least 1 month deposit due to "move out cleaning fees" (regardless if you keep the place spotless).
If you know prices in the bay area, paying 6.5X that rent up front is pretty painful, and not being able to recoup 2.5X your rent is also kind of rough.
I'm lucky that my salary didn't shift too much (aside from currency exchange) when I moved here, but generally salaries are much lower across the board in Japan, even Tokyo.
Food is similarly priced here as SF, depending on what you choose to eat. Some meals can be considered cheap (gyudon, ramen, etc) but you have to remember portion sizes tend to be different and meals don't typically include drinks here. It's also really nice that tipping is not part of Japanese culture, since I personally find it adds a lot to the total cost when you eat out a lot (like my fat ass).
Now, to add a huge caveat to all this, you can live cheaper in Tokyo if you choose to. When I moved here, I chose to keep my lifestyle roughly the same (though I didn't buy another car or motorcycle, so there's some savings). First, if you want to live more cheaply, don't bring a pet. You can also downgrade your lifestyle a fair bit which is an interesting difference from SF. In SF, even a small studio can go for a lot of $$$, but here, you can find small studios for $500-$1500USD relatively easily (whether they accept foreigners or not is a different issue).
TL;DR: Can definitely be cheaper depending on what sacrifices you're willing to make. With no sacrifices, about the same as SF, except up front costs.