"Volumes" are the Docker construct used to store persisted data. You use a volume when you want to decouple the lifecycle of your data with the lifecycle of the app. You can either map the volume to a directory on the host (ie, map /var/lib/mysql to ~/data/mysql), or you can allow Docker to manage it (where it'll live in /var/lib/docker/volumes/blah/blah).
You don't upgrade a running container. Imagine that containers are immutable; to launch a new, upgraded version, you re-build the image in your build/dev environment, and re-launch the image into production. If you're using a volume, you get to use the same backing data.
It's rare that apt-get postinstall will affect any data that you would persist -- app-specific data you'd keep in the image/container, and mysql data for instance you don't really want apt touching anyway. But if a data migration is necessary you'd either manage it with a "utility container" (image that's designed to run a script then stop, rather than run and keep running).
You don't upgrade a running container. Imagine that containers are immutable; to launch a new, upgraded version, you re-build the image in your build/dev environment, and re-launch the image into production. If you're using a volume, you get to use the same backing data.
It's rare that apt-get postinstall will affect any data that you would persist -- app-specific data you'd keep in the image/container, and mysql data for instance you don't really want apt touching anyway. But if a data migration is necessary you'd either manage it with a "utility container" (image that's designed to run a script then stop, rather than run and keep running).