In short: Handle guest-OS patching just like you would in the physical world, there is no need to change the tools or processes just because the OS runs in a VM now.
In the virtualized environment you could take advantage of snapshots in case patching goes wrong, but be careful not to useruse snapshots and remove them in a timely fashion after you confirmed your system runs fine.
To clear some misunderstandings:
I read something like 'put a VM in Maintenance Mode' before the patch install happens,
You put physical ESXi hosts into maintenance mode before installing patches. This has nothing to do with VMs.
Also, in the vCenter Server client, I see "Update Manager" which seems to do some automatic updates on either Host? or VMs? Can anyone tell me what you normally do to apply the patches?
vCenter Update Manager is mainly for patching/upgrading physical ESXi hosts or virtual appliances but it can also upgrade the VMware Tools package of a Guest and VM hardware level; but no guest-OS or any other patches, so this is unrelated as well (it used to have capabilities to manage guest-OS patches in the past but that has been removed in recent releases).