I would personally look at the map. If the line surrounding the CTR is dashed, it is not always active and usually the activation times would correspond to the airport opening times. If it's solid, it's supposed to be active 24/7.
In my experience, if the airport beneath a 24/7 CTR is closed, it is the ATC facility of the surrounding airspace that can give you a clearance to fly through and this will normally be given without hesitation.
In any event, you would have to confirm that the airport/CTR is closed with someone, so I would simply call the ATC facility that surrounds the airspace (some sort of INFO frequency, usually) what the status is. They'll either tell you it's closed and fly straight through, or give you a clearance to fly through.
Not really relevant in the winter, but in the summer, if an airport is closed, they may still use the field for glider launch. Better stay clear of the ATZ in any case.
For a more formal answer you'd have to look it up in the AIP.