I think Ken Love has a youtube video about this. He suggests taking the ice box cooler out to the orchard, and as soon as you pick those babies sticking them into that cooler.
I've had jaboticabas in fridge for many weeks, and still eat them. After one week or so they will start to wrinkle and ferment. But even after starting to ferment, to me they taste like wine, and i eat them. I think it's best to use a bowl, rather a closed container, so they can continue to 'breathe'. You can also ofcourse freeze them whole for long storage.