Technically, yes you are supposed to keep paying but … I retired at 63 and intended to continue payments but the pension office people I talked to suggested that I don’t pay. So I didn’t, and just waited to my 65th birthday to apply.If you are under 65 you still have to pay your national pension contributions - these are set at about ¥20,000 a month and paid through city hall. Of course, once you start getting the Japanese pension you stop paying into this scheme.
As to the ability to continue health insurance through Kyosai for 2 years after retirement, this will almost certainly be the cheaper option for the 1st year of retirement. For those with good post-retirement incomes, it may also be cheaper for your second and final year as well. Enjoy getting a Kyosai subsidy for Ningen Doc while you can! Your employer’s HR should explain health insurance options, as well as retirement bonuses and tax obligations sometime in your final 2~6months. If they don’t, maybe you should request one.