I recently tried to ship an old MacBook Pro and iPhone to my sister in the U.S. We took them to the post office, paid 8,000 yen, and went on our merry way. A postal worker called us later to say that they couldn't ship the items as they contain lithium ion batteries, which fall under the "dangerous goods" category. We asked about shipping by sea, but were told this was not an option.
This then led to researching various other options for shipping these items abroad, although in the end, the cheapest option is to ship just the laptop without the battery in it (it's dead, anyway).
However, we also discovered that although Japan Post says you can't ship electronics with lithium ion batteries overseas, another pdf on their site says that you CAN ship some electronics with lithium ion batteries, but only to certain countries, and only by sea (surface) mail (船便, ふなびん, funabin). Of course, the U.S. is not one of them (which would explain why we couldn't send the laptop and cell phone), but Canada, England, the Philippines, Singapore, South Africa and several other countries are. You can find the pdf here, although it's all in Japanese.
Also, you can send the tiny little wristwatch batteries by airmail through Japan Post.
So what are some other possible options for shipping your lithium-ion-battery-containing electronics from Japan?