Hello Vasil,
Thanks for your questions regarding ArcGIS 10.4.x and Oracle 12.1.0.2. As you mention- yes ArcGIS 10.4.x is supported with Oracle 12.1.0.2: Oracle database requirements for ArcGIS 10.4.x—System Requirements | ArcGIS Desktop
Although not mandatory, we typically advise customers to upgrade database client software in step with their database- so as you mentioned installing the appropriate 32 bit (ArcGIS Desktop) and 64 bit (ArcGIS Pro / ArcGIS Server) Oracle 12.1.0.2 clients on the necessary machines.
Regarding the upgrade itself- we typically recommend performing an upgrade in a test environment using a backup from the production environment to ensure the workflow goes as intended. Once the test environment upgrade is completed- you can then use the same workflow in production. The typical workflow involves taking a full database backup / dump for safeguarding, and then either doing an Oracle in-step upgrade, or restoring your Oracle backup to the new environment if you are going to a new server environment.
Oracle backups—Help | ArcGIS Desktop
Upgrade a geodatabase in Oracle—Help | ArcGIS Desktop
I hope this is helpful for you!