Is "Monitor Process" only available for System Admins?

You need to have access to the designer interface, but you can give this role to non system administrators.

Yes. Anyone who can access the process modeler (System Administrators) can monitor processes.

If the user is not a System Administrator, then they need to be in the Designer Role in order to access the designer interface & process modeler (forum.appian.com/.../User_Roles.html). I believe users in the Designer Role can view a specific process if they have at least 'Viewer' rights, then they should be able to 'Monitor' and view the data (forum.appian.com/.../Configuring_Process_Security.html).