Japan is somewhat notorious (and quite rightly so!) for its strict stance on illegal drugs and lack of clarity on importing necessary medications while traveling from overseas. After encountering several people online attempting to bring their medications into Japan, I decided to investigate the issue further. It was a bit of a slog, but here's what I found out.

Not to bury the lead, let me answer the title question with a qualified, "Probably. That is, usually. Most likely you can bring your prescription. But do you know how?"
So, here's a brief guide on HOW to bring medicines you need to stay healthy during a short visit (less than one month) in Japan. I won't be covering bringing or shipping more than one month of medicine, as that's way, way more complicated and something you'll need to figure out on your own. (But it's definitely possible!)
You can bring any prescription medicine into Japan without any special
procedures on condition that:
(1) you bring/send it only for your own use
(2) it is not any prohibited drug in Japan such as Methamphetamine,
(3) it is not any especially controlled drug in Japan such as Narcotics,
(4) quantity is not more than a one month supply.
(5) Note that it is not permitted to SEND Psychotropic drugs ahead of your arrival. (But they may be carried for own use.)

Many people don't KNOW if their drugs violate #2 or #3 on the above list, so what can you do about it? You can ask your doctor or a pharmacist if it is a Methamphetamine or a Narcotic, but if not, that still doesn't find out if it's on Japan's banned list!
If you don't know if your specific prescription medicine is prohibited, you must email the Pharmaceutical Inspector in your place of arrival’s neighboring office. (emails below).
Be certain to include the following information:
the name of the International Airport where you will arrive (Narita, Haneda, Kansai, Chubu, etc. )
the product name of your medicines
the name and the amount of active ingredients (ex: XXmg / tablet etc.),
the figuration of medicines (is it a “vial” or “tablets” or "capsules" etc.)
the amount of medicines which you need to bring (“one month supply” or "300 tablets,” etc.)
(They have a backlog, and may take up to several WEEKS to reply, I am told!)
The Official Emails (English okay) and Phone Numbers (Japanese only) are as follows:
Place of arrival: Narita International Airport, Haneda International Airport, etc.
(Eastern/Northern Japan area)
Kanto-Shin’etsu Regional Bureau of Health and Welfare
TEL: +81-48-740-0800 / FAX:+81-48-601-1336
Email: yakkan@mhlw.go.jp
Place of arrival: Kansai International Airport, Chubu Centrair, Naha Airport, etc.
(Western/Southern Japan area)
・ Kinki Regional Bureau of Health and Welfare
TEL: +81-6-6942-4096 / FAX:+81-6-6942-2472
Email: kiyakuji@mhlw.go.jp
References: