Most herbs in the ground will take cold temperatures fine, they just stop growing during the coldest months , some die back a bit, most grow again as soon as its warm.
Our garden in Bulgaria had (as far as i know) nothing done for years, apart from being chopped back here and there for photos etc, when clearing it I found rosemary, lavender, mint that were growing fine
Growing in pots is fine if you want to keep eating them during the winter, but i would just grow some pot herbs as well as in the ground they will do much better
Watering is the killer with most pot plants. You need a very free draining compost not just the stuff from the shop add things like bark chips , cat litter or grit mixed in, unlike the usual advice i would say never to add stones to the bottom of pots, drainage material should be right through the mix
If you add stones, broken pot or similar to the bottom of a pot it causes a perched water table to form, the interface between the two is very wet, the roots go rotten in the bit between the two substrates.
Best thing in a pot is 50/50 hard stone type cat litter or gravel the same sort of size, and compost, check there are loads of holes in the pot at the base and the plants will last longer