Greetings for the day!!!
In order to answer your query of only the user (or admin) can update the document, you just need to revoke the access of Public wrtite from that document. After the success of above operation, only user and admin have access to update that document. Please have a look at this link for the code snippet.
P.S In order to update the document, you can use any update methods of Storage service.
Let us know if you find any issue while integrating it.