The Updater plugin automates updates to jEdit. It can update to the latest point-release version (e.g. jEdit 5.5) or to the latest daily build. You can configure the plugin to run automatically on startup, or once in a few days when jEdit is running. It also offers an action to manually trigger an update.
When updating, it shuts down jEdit, and restarts it using the script of your choice (configurable from plugin options).