Yes! (This is new as of the Fall of 2014)
Godaddy just silently or quietly (no announcements that we could find) turned on a one click staging tool for its Managed WordPress Hosting service. It quickly and easily creates a full copy of your live/productions website on a new temporary url that godaddy provides. It took about 4 minutes to create in our initial test where we clicked the button and waited 4 minutes.
The temporary url shows up in a list with the domain that it was copied from. This means that this does not use up your allowed number of sites for your plan. If we understand the implications of this correctly, if you are on a plan that allows 5 sites, that means you would be able to also have 5 staging sites, one for each of them.
The staging tool does allow you to copy the entire site or just the theme and plugins with no content going from live to staging.
We have not yet tested the tool and options pushing things back from staging to live, but there are several options there.
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