I'm not sure why Artemius is called the Great Martyr -- his beheading seems no worse than the beheadings of many contemporaries, including those he persecuted. But combining accounts from a couple of sources, I come up with the following about him.
Artemius was a leading Christian in Alexandria during the reign of Constantine the Great. I don't know how leading he was, since I don't read that he was at any of the big councils or wrote any famous tracts, but he was appointed Governor of Alexandria by the Emperor Constantine. As such, he was a dedicated adherent to the Arian heresy, famous as a persecutor of the proto-Orthodox Christians.
Constantine's grandson, Julian the Apostate, became emperor and attempted to restore polytheism to its privileged place within the empire. Artemius reprimanded the emperor for this relapse into paganism, and for his trouble, he was tortured, decapitated, and eventually canonized as a saint.
The reign of the Julian the Apostate must have been a perilous moment for the whole empire. It is something I'd like to read more about.