Not much in the way of scripts, but the systemd file I use is something like this:

  [Unit]
  Description=App
  After=docker.service
  Requires=docker.service

  [Service]
  TimeoutStartSec=0
  ExecStartPre=/usr/bin/docker pull app
  ExecStartPre=-/usr/bin/docker kill app
  ExecStartPre=-/usr/bin/docker rm app
  ExecStart=/usr/bin/docker run --name app --rm=true -p 80:80 app

  [Install]
  WantedBy=multi-user.target

With something like dokku I could just push the git repo containing the Dockerfile and it would accomplish the same thing

You probably also want to add: PartOf=Docker.service

This will ensure that a restart of the docker service will trigger this service to be restarted.