The Whiteout level is unlocked by first collecting 20 Orologions (little stopwatches). These are pickups that drop from light sources. Once you have gotten 20 Orologions to unlock EXTRA: Whiteout you will unlock the new map ''Whiteout'' too.
If you have played before just load a level and quit out as it will unlock retroactively.
Surviving for 20 minutes itself is really easy, select Queen Sigma (or your character of choice) and select the Mad Grove arcana to immediately collect the bonus items. then focus on getting the other achievements whilst surviving for 20 mins.
Visual guide: