Aegis, our current naval anti-missile defense system, tracks missiles and projects their course and speed. Regardless how fast that missile is traveling, if a ship fires an anti-missile missile that will intercept the missile along it’s project flight path, it will shoot the missile down. If the hyper-sonic missile is fired close enough that there is not enough reaction time than countermeasures include the AN-SLQ-32 to jam the missile, the phalanx system to track and shoot-down the missile, as well as Chaff.
If an extended range Standard missile (SM-6) can intercept a ballistic missile than it can intercept a hypersonic missile. Remember Ukraine shoot down Russian hypersonic missiles using the Patriot system.
No it can’t shoot down a hypersonic cruise missile. Not with its current capabilities.
The story about the Patriot shooting down the Kinzhal missiles is YouTube/Business Insider propaganda garbage.