There really isn't a direct staging -> production approach with regards to the underlying data. Most clients do one of the following:
Copy their live site to a test install, do the work, then move that test install back over to the live site entirely.
Copy your live site to a test install, set it up how you want it. When you are ready to update production, make the same changes there manually. You can export some things, such as themes, language packs, pages databases, etc. to facilitate this process.
Unfortunately there really isn't an out of the box solution for copying setting values or menu configurations over.