What kind of contest had the Old Man participated in that he remembers!?

That would be arm wrestling.

As the sun set he remembered, to give himself more confidence, the time in the tavern at Casablanca when he had played the hand game with the great negro from Cienfuegos who was the strongest man on the docks. They had gone one day and one night with their elbows on a chalk line on the table and their forearms straight up and their hands gripped tight. Each one was trying to
force the other’s hand down onto the table.