So, in general: as long as you're travelling for less than a month, you're allowed to bring, without any special permission:
- 1 month worth of prescription medication, including injectable drugs (only pre-filled or self-injection kit)
- 2 months worth of non-prescription medication, including disposable contact lenses,
So what is restricted or prohibited? Taken from the Consulate General of Japan in Seattle page, the restriction seems to be on stimulants (e.g. Adderall, methamphetamines, amphetamines) and highly controlled medications (e.g. morphine, oxycodone, pseudoephedrine, codeine, fentanyl; source from the Seattle embassy as well as the Australian embassy). Opium and cannabis are prohibited from being brought to Japan, even if you get them legally from your home country for medical purposes. If you take any prescription-grade pain killers, I would also check if there are any restrictions. If you're just taking medications for things like high blood pressure or high cholesterol, you're likely in the clear.
Travelling for more than a month? If you'll be in Japan for more than a month and need prescription medications, you should fill out an import certificate (Yakkan Shoumei: more info here).
On psychotropic medications, e.g. antidepressants: The Seattle embassy states that advance approval isn't required as long as you're under the quantity limits (listed here, on page 8-9); if you're bringing in more than this, you'll have to apply for permission via the form in the link on page 4. Like I mentioned above, stimulants are restricted.
Do you need to bring a prescription? I've seen some suggest bringing the prescription to Japan, which is probably the safest thing to do; but I think if you bring the medication in the original container with the name, it doesn't contain any restricted substances, and it's clear you're not bringing more than a month's supply, it's probably not necessary. Again, if it's something common like medication for diabetes, high blood pressure, or cholesterol, they probably won't look twice.
Regarding over the counter medications: If you're planning on bringing over the counter medications, I would check if they contain prohibited substances like codeine. I did find a site that lists some apparently restricted over the counter medication, though I'm not sure how recent/accurate the list is. People seem to warn against bringing Vicks inhalers and Tylenol Cold because they contain some restricted drugs. Personally, I don't think over the counter medications need to be brought to Japan since Japanese pharmacies seem to be well-stocked, but I haven't gotten sick in Japan beyond a minor cold.
Finally, if you want to contact someone to ask if a certain medication can be brought into Japan, there is contact information listed at the bottom of this page. If you are concerned about a certain medication you take, it's better to check beforehand!
The only medication I would bring to Japan other than my regular prescription medications would be some oral antibiotics, since those are not available over the counter but infections (e.g. bladder infections) may happen for one reason or another. If you're prone to such infections, that may be something to keep in mind. If you have any experiences regarding bringing medicine into Japan you'd like to share, please leave a comment!