Please, Dennis, and everyone else who reads this message, thanks for the kind words, but I am certainly no expert on these or any matters. I just thoroughly enjoy the study of the Naval actions in the war, and I am still learning a lot, as I slowly indulge in much research in this area. I have many publications and even copies of first hand accounts in my personal library, and often have to peruse these for the data therein, but it is purely a labor of love. These items, in my collection, can often provide much data on the Navies, some of which I cannot find elsewhere, but I try to assist those who seek information, when I know I have this at hand.
Anyway, the CUMBERLAND, being a blockade runner, would not have had the initials CSS appended to its name, unless it was regularly commissioned into the Confederate States Navy. According to volume 17 of the ORN set, it was actually stated to be fitting out as a privateer at the time of her capture. I assume it would have been taken to Key West, since it was the closest base of operations for the vessels in that Gulf Blockading Squadron. It would pay to check the deck logs of the DE SOTO for any details at all. [Page 644, volume 17 of the ORN actually names several persons who were indicated to be on the CUMBERLAND - Daniel Graham, chief engineer Armstrong, Daniel Golding, Luke Shaw and David Pennel.]
As I have indicated before, to other researchers, it does tend to get rather confusing, in relation to personnel of the blockade runners and even the privateers, as some of these personnel were also members of the Confederate States Navy, though often they were private individuals who manned the blockade runners purely for financial gain. And yes, if they were captured, they were often imprisoned as regular prisoners of war, though I have also come across many cases of British or other citizens, aboard these blockade runners, who were released because of their status.
I have come across deck logs of a number of Union Navy vessels, in which prisoners of war were taken aboard or transferred to other vessels or North for imprisonment. Sometimes they were named, and sometimes they remained nameless, so it would be quite hard to tell if your man would be named on the vessel he was transported in. I guess the only way to find out is to be prepared to take the time to peruse these entries in the deck logs. No doubt you will sometimes find some other, non-related entries in the deck logs, but they can be a great source of interesting data, anyway, no matter whether you locate the data you seek, or not.
Perhaps you might want to peruse other sources of data for the name of your Confederate, such as Confederate pension records, or his military service records, as they may provide some additional data, though often Confederate records are rather sparse. If he did get a state pension, he may have given some details of his capture, and where he was sent to.