Interesting question. I just looked around on some online forums, and the answer seems to be "if it doesn't have a name, the Japanese post office will not deliver it". And the writers cite a Japanese postal law wherein the Japan Post Office has some obligation to make sure the intended recipient is indicated on the letter, so that it can be delivered to that person. So on a residential letter, if the intended recipient is unclear, the letter gets kicked out of the system. (Letters to a business can get addressed to the company or department, no private name needed).
Other posters say, "yes that is the law, but sometimes stuff still gets through without a name". So I suspect there is a slim chance it could get through if you write "current resident" or something as the name. My gut feeling is; if your target address is in Tokyo, and the post office is used to handling large volumes of mail from overseas, it might get through. If you are sending it to some place out in the provinces, where any "out-of-the-ordinary" mail is subject to scrutiny, it will probably get kicked back.
Current Resident
4-5-6 Marunouchi
Tokyo, Japan 123-0000