Claudius has a bust in his room, which resembles emperor Lucius Verus, who co-ruled with Marcus Aurelius 150 years later. Or at least a man with a heavy full beard which first became fashion among noble Romans in the second century AD.
Augustus addresses the poet as Horace, and one may assume this is the famous Roman orator who gave the Classical World his Odes, and such. However, Horace died in 8BC, about sixteen years before the setting of this time. The poet Ovid whom Augustus also mentions was indeed still alive for the next eight years.
Claudius is depicted as being upset by the gladiatorial combat. In fact, he was an enthusiastic fan of blood sports. But this programme is not history but an adaptation of Robert Graves' novel, and that specifically depicts young Claudius as being very squeamish about death in the arena, but becoming an enthusiast in his fifties when Emperor.