Generally... not as quickly. The salt content of the wine is much higher than drinking wine, which will preserve it longer. It will still "go bad" eventually. (Going bad is more a decrease in quality. It will still be edible, but unpleasant. It won't make you sick, except by the bad taste!)
See this from a Wikipedia Article: "Cooking wine typically available in North America is treated with salt as a preservative and food coloring."
So, it's safer than normal wine, but I'd try and use it soon. The sooner, the better.
Water freezes at 0° C (or 32° F), but ethanol freezes at -114° C, so you can guess your vermouth with freeze somewhere in between those two temperatures.
Noilly Prat is 18% ABV, or 15% alcohol by weight, which means it would freeze at around -6° C to -7° C, or 19-21° F. Your freezer may or may not be tuned to be that cold, but if not, you should be able to crank it a bit and get your Noilly Prat just frozen.
That said, I would not really recommend leaving an ice cube tray full of vermouth hanging out in your freezer. You may notice that old ice tastes funny. Even though you may not notice your freezer smelling, that's partly because the cold blocks smells. Just because you don't notice them doesn't mean the odors aren't there to ruin your Noilly Prat. I would recommend leaving them for a day until just frozen (covered if you like), and then moving them to a Ziploc for long term storage.
All that said, freezing will not halt oxidation altogether, but it will help, as only the exposed surfaces areas should be able to be oxidized. Since ice cubes by design have a relatively high surface area, you may consider minimizing the exposure to oxygen by other means.
If you have a vacuum sealer, using that would be a fairly simple means to limit that exposure, but a similar effect can be done with a simple Ziploc bag, as described here (note this link is focused on using this technique for cooking, but it is translatable to your scenario). Drop your ice cubes in a bag, and then slowly lower the bag into a container of cold water (the colder the better, here, to minimize melting), leaving the seal exposed. I sometimes clip things onto the bag to weigh it down as well. Then begin closing the seal, and lower the bag further into the water as you continue sealing, so that only the unsealed portion is exposed. Finally complete the seal and you should have a reasonably low-oxygen environment to mitigate further degradation of your vermouth.
Best Answer
I tried freezing leftover wine many years ago, and it does work. You can just freeze it in the original bottle.
The flavors are somewhat muted compared to the same wine unfrozen. Not recommended for drinking the wine, but it is acceptable for cooking purposes.