Upgrade RDS instance with zero (minimal) downtime

If this is RDS multi-AZ, the impact is minimal. From Will a Multi-AZ deployment help reduce downtime during an Amazon RDS MySQL modification?

Because an instance class change requires a newly defined set of hardware, this change isn't an online operation, and therefore requires downtime. A Multi-AZ deployment of an Amazon RDS MySQL DB instance can significantly reduce any impact. This is because the update doesn't happen simultaneously to the primary and standby. The standby instance is first modified, causing a failover. After the failover, the new standby is modified. The downtime required includes the duration of a failover completion, typically 60-120 seconds, and completion of the DB engine's crash recovery.

You can modify RDS from single-AZ to multi-AZ and vice versa as per What's the impact of modifying my Single-AZ Amazon RDS instance to a Multi-AZ instance and vice versa?

When you change your Single-AZ instance to Multi-AZ, you don't experience any downtime on the instance.

When you change your instance from Multi-AZ to Single-AZ, you don't experience downtime on the instance.