Hi,
If I am not mistaken you should be able to backup Context configurations from the Context itself through CLI with the command "copy running-config <destination>".
I would imagine though that for copying the System Context configurations and all the different files on the ASA you would need to attach the "admin" Context to some sort of Management network you have. When you want to copy different files from the ASA through the System Context it will need the "admin" Context connectivity to reach the server(s) to/from which you copy information. To my understanding you would be copying the files through the CLI of System Context and not the "admin" Context.
I can't really comment on the ASDM side of things in Multiple Context Mode as I barely use it. So I am not up to date on what is supported through ASDM when the ASA is in that mode.
But I would imagine that the "admin" Contexts management network connection would be a good solution when managing the ASAs files and in general to manage the ASA.
- Jouni