My curiosity got the better of me and I went digging concerning what would happen if you caught and threw a shark in a freshwater... and I found out it could actually survive for a period of a few hours to a few days.
The way osmoregulation works in sharks, it does not allow for switching between salt and fresh water unless the fish in question has adaptations to allow this. Some sharks, like the Bull shark, can adapt to brackish and fresh water but the Great White cannot do that. When moved to a freshwater environment, the osmotic pressure would cause water to enter their bodies, leading to a condition called osmotic stress. This can result in severe physiological issues and ultimately death.
How they would transport sharks from boats/sea to the Colosseum is another matter though.