The 29-year-old Roman's home was searched by police and he is being investigated for inciting racial hatred.
Police said they identified the unnamed man after examining the three parcels, witness statements and from surveillance of far-right activists.
The man has past ties to the Italian neofascist Forza Nuova movement, police aid.
In a speech to mark International Holocaust Remembrance Day on Monday, Italy's president Giorgio Napolitano condemned the anti-Semitic threats as "a miserable provocation".
The threats drew criticism from the Israeli government and representatives of Italy's Jewish communities as well as Rome's mayor Ignazio Marino and other politicians including integration minister Cecile Kyenge.
The threats prompted authorities to boost security in Rome in key areas including the synagogue where events marking Holocaust Remembrance Day were taking place.