Do I need to have an app store installed in order to get Tor updates on Android, or will Tor download updates automatically by himself (like Tor Browser on Desktop does)?
Tor browser for android can be updated automatically as well as manually. Updating Tor Browser for Android automatically assumes that you have either Google Play or F-droid installed on your mobile device. After installing google play, you click on the hamburger menu next to the search bar and navigate to "My apps & games" > "Updates". If you find Tor Browser on the list of apps which need updating, select it and click the "Update" button. While on F-droid, click on "Settings", then go to "Manage installed apps". On the next screen, select Tor Browser and finally click on the "Update" button.
In order to update Tor Browser for Android manually visit the Tor Project website and download a copy of the latest Tor Browser release, then install it as before. In most cases, this latest version of Tor Browser will install over the older version, thereby upgrading the browser. If doing this fails to update the browser, you may have to uninstall Tor Browser before reinstalling it.
To learn how to install F-droid and more information on the Tor browser for Android, you can refer to this link: https://tb-manual.torproject.org/mobile-tor/