You can, but you should not.
A paid internship is much more likely to be worthwhile than an unpaid one. Being paid ensures that you have internal visibility. You're paid, so you're in the system. Since the company is spending money on you, you must be worth something. An unpaid intern is nobody and needs to justify themselves for everything: to get an office or desk, to get a computer and a phone line, to get access to buildings and documents, to get a budget for projects or travel, etc. If you're paid, no matter how little, all of these things are much more likely to go smoothly. Even if you have a supervisor who cares about you, the rest of the company needs to know they need to care too.
Since you'll be at a big company, especially an international one, they will have people who know about visas and can get matters resolved quickly (often by paying a firm that specializes in the corporate visa business). Don't be daunted by the prospect of administrative difficulties: you'll probably have to fill in a lot of forms, but you shouldn't have to figure out which are the right forms by yourself.
If you think your visa won't be approved in time, you should let your supervisor know about the risk, and keep them informed about your projected start date as you handle the paperwork with the company's HR, with the visa contractor if any and with the government. Usually internship dates are mainly constrained by your study institution — it needs to start after the lectures finish and before the awarding of the degree. If the start date is really a problem, you may start by working without being paid, if that's financially acceptable for you, but make sure that you're already in the company's books with a future paid period duly entered in the company's databases.
If being paid is really an insurmountable problem for administrative reasons going beyond the delay to obtain a visa, then try to arrange for the company to budget something for you, for example to pay for your housing. Even if you don't need the money, being on the company budget is important.
You sound like you're in Japan already, so you probably know, but keep in mind that Tokyo is a pretty expensive city. What is a decent living income elsewhere (even in Japan) might not pay for housing there.