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Yes, there are some hotels at the airport and elsewhere on Goverrnor's Island but actually you're not saving a lot of time. Because of the road network Copacabana is only 30-40 minutes away, and Icarai about the same (though geographically nearer). The airport island isn't so attractive, whereas the beach is of course well known. Personally I'd be OK with a good quality hotel on Copa, if you allow an hour for the transfer you would be more than fine. I think the crews stay in that general area too.
I think you can put the assistance request on MMB.