- How to use rsync over USB on Android with adb
- the rsync man page for further details on available parameters and possibilities
- How to rsync to android (in case you want to advance this to include your Linux machine ;)
RSync is also available on other operating systems, see e.g. this list on Wikipedia - there's another RSync app on Playstore: Acrosync – but it wasn't updated since 2013 and neither is it well rated (2.7 stars at 82 votes)
- Backup your Android phone with Rsync describes a similar approach as above, involving different resources
- How to automate command line when specific application is launched? (for on-demand syncing; notnote that e.g. tasker also allows to create widgets for tasks, which then can be "tapped on demand" to execute it)