In this post, I'd like to give you the main lines of upgrading and patching Cloud Control
UPGRADING
Of course, your situation may be different. But if you have a simple Cloud Control setup, you may work with the following guidelines.
You should be working with Cloud Control 13c, specifically with 13.5
If you are on a lower version, first upgrade it to 13.5. I would suggest that 13.4 is fine as well, but an upgrade could be made nevertheless.
If you need to upgrade, download the 13.5 version via edelivery (wget is preferred to do so).
When you start it up, you will get the option to upgrade. An example of the steps is on:
https://eclipsys.ca/oracle-enterprise-manager-upgrade-from-13-4-to-13-5/
This will create a whole new tree structure for gc_inst and OMS-home folders. So you need a lot of space.
You can follow all steps in the upgrade guide:
https://docs.oracle.com/en/enterprise-manager/cloud-control/enterprise-manager-cloud-control/13.5/emupg/introduction-upgrading-enterprise-manager-system-13c.html
After the installation, you can remove the old directory structures, but be careful with that.
https://docs.oracle.com/en/enterprise-manager/cloud-control/enterprise-manager-cloud-control/13.4/emupg/deleting-old-oms-home.html
After the upgrade, you will need to upgrade all the agents.
Start with the Central Agent (on the OMS host itself). You can easily upgrade agents with the cloud control option under Setup>Manage Cloud Control>Upgrade Agents
After the central agent, you need to upgrade all target agents. You can do a few/some/a lot in one go.
Note that upgrade agents requires a root.sh script to be run.
Note that after the upgrade of the agent you should run post-agent-upgrade steps (which cleans the old agent home).
Upgrading agents is a very smooth process that will not require any downtime (just briefly of the agent itself).
PATCHING
Now the patch áfter the 13.5 upgrade. Because you now have 13.5.0.0.0 installed on OMS and all the agents.
There is a monthly patch, which can patch the OMS to 13.5.0.10.0 for example. This fixes a lot of bugs and issues.
You can find the patch in:
https://support.oracle.com/epmos/faces/DocumentDisplay?id=2902989.1
After downloading the latest patch (wget preferred), you should also download the latest OMSPatcher tool (follow the readme of OMS patch).
And you should also create a property file, which helps you installing the patch.
With the property file created and OMSPatcher tool upgraded, you can now
omspatcher -apply -analyze -property_file /pathto/mypropertyfile.txt
emctl stop oms
omspatcher -apply -property_file /pathto/mypropertyfile.txt
The patch is in-place and gets you to OMS 13.5.0.x.0 (last patch).
Hope this helps you in your patching. Always follow the steps in Oracle Support documents!