You should always keep sauerkraut refrigerated if not using it within an hour.
If sauerkraut is left out longer than an hour it could start growing bacteria and to keep the flavor of the sauerkraut from changing.
Once you open a jar of sauerkraut it must always be refrigerated if not using the opened jar of sauerkraut within an hour.
If you leave the sauerkraut out overnight it will grow bacteria and be spoiled and will have to be thrown out.
But if you keep the sauerkraut refrigerated it will last for a week or more as long as you keep the jar of sauerkraut sealed with the lid and keep it refrigerated when not using it.