Hi,
first of all, it shouldn't be a common problem, since using NAT between vCenter and ESXi hosts is an unsupported configuration by VMware 
How it should work anyway:
In vSphere Client go to Administration -> vCenter Server Settings -> Runtime Setting and set your public IP for your NAT network as "vCenter Server Managed IP". After that this IP should show up in the Update Manager patch store drop down menu.
Good luck!
Regards