I was in a similar position to you in tracking down an ancestor from Copenhagen, although I did have his "Danish" name from a family Bible record in addition to the birth date.
There is not a lot of name variation in Denmark from that time, although Copenhagen is an exception (at least from what I've seen in my own family). German names were more common in Copenhagen than in rural areas. I'd guess that his last name is "Jensen." James could also be a form of "Jens." Henry might be the German "Heinrich" which was not uncommon in Copenhagen. You do also see English names sometimes in Copenhagen, so he might have been James Henry Johnson!
I'd suggest going through the church books (http://www.sa.dk/ao/SoegeSider/Kirkeboeger.aspx
) and looking for someone with a name similar to his born on or around the date you have. Select "Copenhagen" from the top drop-down and then look through the books. I'd start with Vor Frelser and Trinitatis, as those are two of the largest churches active in that time period. (When I did this for my ancestor, I went in alphabetical order and found him in Vor Frelser!)
You might also want to track down any more information you can find on his parents and brothers to narrow things down. Unfortunately Copenhagen is large and "brute force" searches can take a long time!