The man was caught after allegedly giving some of the drug to a customer, who fled amid the confusion, Ansa reported.
Police said the man was carrying 140 doses of cocaine. They also recovered a stolen wedding ring and luncheon vouchers.
More than one million people have booked to see the Shroud, which will remain on display in Turin's Cathedral of St John the Baptist until June 24th.
Viewers will be afforded only a few minutes each in front of the original although they will be able to linger longer in front of a specially-made model and a related exhibition.
When the shroud was last presented to the public, in 2010, more than two million people filed past it.