docker export -o <nameOfContainer>.tar <nameOfContainer>
Might need to prune the existing container using docker prune ...
Import with required modifications:
cat <nameOfContainer>.tar | docker import -c "ENTRYPOINT service mysql start && /bin/bash" - <nameOfContainer>
Run the container for example with always restart option to make sure it will auto resume after host/daemon recycle:
docker run -d -t -i --restart always --name <nameOfContainer> <nameOfContainer> /bin/bash
Side note: In my opinion reasonable is to start only cron service leaving container as clean as possible then just modify crontab or cron.hourly, .daily etc... with corresponding checkup/monitoring scripts. Reason is You rely only on one daemon and in case of changes it is easier with ansible or puppet to redistribute cron scripts instead of track services that start at boot.