] He self describes himself as more of an antique Roman than a Dane. By this I think he means exactly how he acts, if that makes any sense. He is sensible and will only believe his own senses, as when he first sees the ghost.

While I believe that Horatio is a character of honest and forthright speech and action, I don't believe that's what he means when he says he's more an antique Roman than a Dane.

He is talking about killing himself. Romans were famous for falling on their swords when they faced dishonor--they thought suicide was more honorable than living with defeat. At the moment Hamlet says he's dying, Horatio feels that he has failed in his quest to help Hamlet restore order to Denmark and that life is not living without his friend.

Hamlet asks him to absent himself from felecity a while and tell the story.