I believe the Elite you are referring to is the one the Master Chief fights in the control room of the Ascendant Justice. This takes place on pages 91-96 of the paperback edition of Halo: First Strike. This is the only detailed 1-v-1 fight in the novel, so it must be what you are thinking of. He is never named.
An Elite in jet-black armor materialized from the wall display, its light-bending camouflage dissolving. It strode toward the Chief, roaring a challenge.
The Chief forced the alien back, step by step. The deck was slippery with blood.Finally he twisted the Elite to the right and launched a powerful open-handed strike into the alien's wounded chest.
The Elite howled in pain and flew back, through the open hatch of an escape pod.
"Get off this ship," the Chief said. He hit a control stud and the hatch slammed shut and the hatch slammed shut. There was a sharp, metallic bang as the locking clamps released. The pod screamed away from the hull.
However, when the Master Chief takes the ship back to Reach, Cortana discovers that the ship is supposed to be commanded by someone called The Guardian of the Luminous Key. While this Guardian is never specifically identified, is likely a Prophet.
It is most likely that this being was the Minor Prophet in command of the Ascendant Justice during the events of Halo: The Flood. Upon failing to make contact with High Charity, this Prophet took tactical control of the Covenant fleet at Alpha Halo, making decisions against the wishes of Thel 'Vadamee.
The Arbiter was definitely not fought by the Master Chief on board the Ascendant Justice, as he was (then as Thel 'Vadamee) commanding the Seeker of Truth and was busy dealing with the Flood menace and the destruction of Installation 04.