well here are the pros and cons.
if you did it apart, you wouldn't have space problems whem trying to do all that, so you wouldn't have to learn brush kung foo, but you would also have to pay attention to your brush strokes, since you'd want to keep them all in the same direction, less you make one side look funny.
if you did it together, well...i can't think of an obvious pro, but for a con, you'd have to be real careful around the sides you'd already done, or even the ones you hadn't started, so that you wouldn't unwittingly drip or splatter and make a funny spot.
also, will you have screws or nails showing after it is assembles, because that is the main reason you do furniture when it is assembles, so that you can fill in the holes and stain them too.
as for me, if i was brushing it on, i'd do it before assembly, and if i was spraying it on with a HVLP, i'd do it afterwards, cause i'd be worried about drips coming down from the control panel parts. also, if you do it before, you might still need to touch it up afterwards due to scratches during assembly. good luck.