How would that work?
Starship uses cryogenic fuel, and thus isn't something you can keep on active standby for years at a time, ready to be launched with a few minutes notice. It takes hours for a Starship to be loaded with liquid oxygen and prepared for launch. By the time you finished getting it ready for launch, it would have been nuked by the other side's second wave.
The only way Starship could be practically useful for missile defense is by launching some kind of defense system stationed in orbit.