Are you talking about upgrades of the host OS or the base of the image? I think you are talking about the latter. Others covered updating the host.

Upgrades of the Docker image are done by pushing a new image, and updating the Deployment to use the new image, and applying it. Kubernetes will start new containers for the new image, and when they are running, kill off the old containers. There should be no interruption. It isn't any different than normal deploy.