This process is partially handled by Japan Automobile Federation (JAF). JAF has their own instructions for their part, but we will summarize the steps since you are already here.
Compared to other processes, this process allows you to skip:
the many hours of practice driving
the many hours of classroom lectures
the written and technical tests for the provisional driver's license
You must have an address in Japan in Japan, this process cannot be started outside Japan
The address on your forms much match the residence card information
Be as polite as possible at the driving center, it's frustrating for them to work with unprepared people in a foreign language
Prepare the necessary documentation
Download, print out, and fill out the application for "Japanese Translation of foreign driver’s license"
Your drivers license, a photocopy is also acceptable but not encouraged
A photocopy of you residence card (在留カード)
¥4000
Using Japan Post Registered Mail, send the items from Step 1 to your local JAF office.
Wait for a few weeks
Receive your translation by Japan Post
Do not miss your delivery, especially if you sent your original driver's license as JAF will not hold onto these if they are returned by Japan Post
Travel to your local driving center
Tell them you are transferring a foreign license (外免切替).
They will send you to pay some fees and get some paperwork
You will then undergo the aptitude test which will:
Test your visual acuity
Test your ability to differentiate colors (red, yellow, green)
Test your hearing
You will take a brief written test
These questions are almost all common sense about safety
You might be asked to undergo a brief driving test on the course at the driving center
If you did everything right, you will get a driving license in Japan!