No, not really. It is not really important to know anything about him. All we really know is that he was a person and that he stood in the same place that the speaker is standing now.
It is not important to know much about this Roman. He is only there to show us that the speaker's feelings are not unique to him. The Roman is there to show us that people have the same problems today that they had over 1000 years ago when the Roman was standing in this spot in England.
So all we really know (or need to know) is that the Roman was human and all humans are essentially the same.
The Roman was of course from a conquering and occupying force of people. We do not learn any specifics about what troubled the ancient Roman, nor do we learn any specifics of the troubles of the modern speaker. (“Then ’twas the Roman, now ’tis I,” line 16). What we are told is that ancient and modern people have problems that are identical to those of past people. We learn that both the Roman and the speaker are alike in their sharing (a) massively powerful impulses that cannot be resisted, (b) thoughts that hurt, and (c) trouble. Such afflictions were, and are, the human lot.