From my experience, no. You will have to setup content staging on each individual server. In the past, I've updated staging and test with a copy of live before I start a major overhaul. In this case, I've documented what needs to be updated from a license, url alias, web farm servers, content staging, scheduled tasks, etc. I've found this backwards promotion to work very well. You might check to ensure all your Site Manager>Settings are using specific site settings vs. inheriting the global settings. This has bit me in the past. I've set global settings and assumed all would be good, which they were until I imported into an new install/instance and those global settings were different or not set.