I don't see how you can cool your NAS without diddling with the case, unless you're prepared to introduce some airflow. If you do add a fan or similar contraption to the box, you can easily filter the air using the stuff sold as an oil filter for cooker extractor hoods, for example.
I did passive cooling a while back on some drives, adding formed aluminum heatsinks which gripped the drives on 3 sides and turned them into true 5" units. Messing around showed that very little passive heat sinking made a big difference. I guess that's no use here unless you're prepared to lose the NAS case.
Since you aren't, you need some kind of environmental heat transport. If you don't want to add a fan to the box, I might suggest a tall vertical black pipe on the top with a filtered opening near the bottom, to get filtered convection running. You can protect the top in a bazillion ways. And cover the box with silver foil.