[ranger207]

Most ballistic missile defense (BMD) currently deployed is designed for a shootdown in the terminal phase, ie after the missile has reentered the atmosphere. Destroying a missile in the boost phase, before it leaves the atmosphere, is the best bet, but that requires advanced intelligence of when and where a launch will happen. If I recall correctly, there's currently no good midcourse BMD for when the missile is out of the atmosphere. Aegis has some midcourse capability, but it's for lower and shorter range missiles, I think. That just leaves terminal defense, like THAAD and the Aegis system, and terminal defense is the hardest phase. So there's no real good option to shoot it down, especially if it turns out your interceptor has a low probability of kill (pk). If a missile was actually coming down into Japan, you can be sure that there will be several interceptors on their way.

As for tech, Japan mostly uses the same tech as the US, including THAAD, which is a ground-based radar and launcher, the Aegis system with the Standard Missile-3 (SM-3) which is a ship-based radar and launcher, and finally the Patriot system of ground-based radar and launcher. However, the Patriot's PAC-2 and -3 missiles are designed for missiles with a shorter range than ICBMs. All of them have data-sharing, so you can use an Aegis radar to track a missile for a THAAD battery to launch.

If you try to shoot a North Korean missile and miss, then either morale goes out the window if it was an NK test, or you have to fall back on the next line of defense (Aegis -> THAAD -> Patriot) and hope that it hits. The further the ICBM goes though, the lower pk you have since it's going faster and potentially has better maneuverability.