Do I need travel insurance and NHI?

I just found out that foreigners residing in Japan for longer than 3 months must apply for the national health insurance. So two questions;

1. Should I buy travel insurance in my home country since I am already going to get health insurance in Japan?

2. Do I apply for the NHI as soon as I arrive or only after I've lived there for 3 months?

Just for reference, I am an Australian travelling to Japan on the working holiday visa in November. I intend to stay in Japan for 10 months.

Any replies would be appreciated.
Thanks!

NHI covers a part of the medical fee only.
Many travel insurances cover rescue expenses, damage/loss of your belongings, liability for damage to others and so on besides medical treatment (I'm referring to common travel insurances sold by Japanese insurance companies).
You should think these two are totally different.

As a Japanese residing in Japan, I have NHI, private health insurance, personal liability insurance, and fire insurance for my house all apart.

On a visa allowing residence you would probably do the NHI sign-up at the same time as registering your address at your local city-hall (which you have to do within two weeks anyway).

Travel insurance is a different product - covers different things. As someone living in Japan you might not find it useful other than the brief time you are travelling to/from Japan (not 10 months which would be very expensive).