Youssef Lamei, pictured above on the right with his son, had his throat slashed twice by an attacker in the early hours of Tuesday morning.
Egyptian police say they have arrested Adel Soliman, 48, after finding him in a disused building.
Officials said they had yet to establish a motive for the killing.
Footage of a man calmly approaching Mr Lamei from behind and slashing his throat twice with a knife was posted online.
The Christian ran an off licence in the predominantly Muslim and people selling alcohol have face attack from extremists Muslims in the past.
Tony Youssef, the dead man's son, told World Watch Monitor the attacker shouted "Allahu akbar [Allah is the greatest]!" and "Oh, kafirs [unbelievers]!" during the attack.
"We then chased him to catch him, but he wielded his large knife in our face and quickly got into a car that was waiting for him," he added.
Youssef, said he believed his father's murder was carried out by a "professional" killer.
"He took less than half a minute, he knew what he was doing, he stood behind my father, quickly pulled the knife from the folds of his clothes, cut the vein of my father's neck two times with his knife to make sure that the vein was cut, and fled."
He said he believed his father had been targeted "because he was a Christian". He said many shops in Alexandria sold alcohol. "There is a shop nearby that sells alcohol and is owned by a Muslim man. Why they didn't kill this man as well?"
"My father was a very kind and respected man and everybody loved him; he had no enemies," he added.
