[Nyororin]

Your medical insurance will kick in here as soon as you go to the local town hall and fill out the forms. You don`t need to worry about when you`ll have insurance.
You will have no income from the previous year to base the payments on, so there is nothing you will need to present to enter national health insurance other than your foreigner registration card - something else you will need to apply for right away. And you are legally required to enter the health plan, so will likely be encouraged to do so immediately when you apply for or pick up your card.

I do not know what medications you require, so can`t comment on how difficult they would be to obtain - but unless it`s something of a highly guarded nature (pain killers, sleeping pills, ADHD medications, ie. things likely to be abused) you should be able to bring over a minimal amount (30 days, or about) without paperwork and then get a prescription inside Japan with little difficulty. If it IS something that is very tightly controlled AND unavailable in Japan, then you really should have started working on this a lot earlier...

In experience with others - it seems that bringing something from your doctor listing your condition and the medication regimen you are on, having that verified by a doctor here (medical check) will almost immediately get you the same prescription in Japan. Bringing over your medical history is really almost a waste of time and effort (unless you have a very very rare condition that no one will have ever heard of) because they will still insist on doing an exam and confirming it. My aunt was even able to get pain medication for her chronic back pain on a tourist visit by going to the doctor and showing him the pills she had brought with her (although only a minimal amount to cover the days that had been unplanned for because of a flight cancellation).

It would have been best if you`d arranged to bring over all the medication you need much earlier on, but I don`t think you`ll need to worry about insurance and getting a prescription inside Japan.