It turns out that U.S. federal law has a provision allowing for arrest prior to an extradition request. Here's the legal language:
18 U.S. Code § 3187 - Provisional arrest and detention within extraterritorial jurisdiction


The provisional arrest and detention of a fugitive, under sections 3042 and 3183 of this title, in advance of the presentation of formal proofs, may be obtained by telegraph upon the request of the authority competent to request the surrender of such fugitive addressed to the authority competent to grant such surrender. Such request shall be accompanied by an express statement that a warrant for the fugitive’s arrest has been issued within the jurisdiction of the authority making such request charging the fugitive with the commission of the crime for which his/her extradition is sought to be obtained.
No person shall be held in custody under telegraphic request by virtue of this section for more than ninety days.
(June 25, 1948, ch. 645, 62 Stat. 824.)
Funny, that bit about using a telegraph! (Obviously, cuz the law was codified almost 70 years ago.) I assume the idea behind this is that people who learn that they have been convicted abroad might try to evade arrest in the U.S. by hiding or leaving the country, and a rapid arrest would put a stop to that.
I have no idea if the Italian authorities will make such a request. But I did read that Raffaele Sollecito was followed to his home in Bari by six policemen, and I presume this is to keep him in sight so that if the Italian Supreme Court upholds the guilty conviction he can be arrested immediately.