Keeping AWS RDS Instances up to date
Your assumption is correct: you will not need to perform patching on an RDS instance, AWS handles that for you. There is a caveat, though: the majority of updates are not forced by AWS, but require an automated maintenance window to execute.

Periodically, Amazon RDS performs maintenance on Amazon RDS resources. Maintenance most often involves updates to the DB instance's underlying hardware, underlying operating system (OS), or database engine version. Updates to the operating system most often occur for security issues and should be done as soon as possible.


Some maintenance items require that Amazon RDS take your DB instance offline for a short time. Maintenance items that require a resource to be offline include required operating system or database patching. Required patching is automatically scheduled only for patches that are related to security and instance reliability. Such patching occurs infrequently (typically once every few months) and seldom requires more than a fraction of your maintenance window.

Source: Maintaining a DB Instance
That document has a lot of useful information about the types of maintenance, the maintenance windows, and the manual actions you can take.
Database engines need patching, too, and AWS refers to this as a minor engine upgrade. When necessary, these upgrades happen during the aforementioned maintenance windows, but you need to ensure your RDS instance has Auto Minor Version Upgrade enabled to take advantage of this feature.
Source: Automatically Upgrading a Minor Engine Version