The penalty in this case
What, for example, is the penalty for what this person is accused of doing?
This (or at least an upper limit) is given in the very article you cite:
At the time of his arrest the Vatican said that if convicted he could face up to 12 years in jail.
There's your answer to that question.
The laws of the Vatican - and a good reference
I don't want to embark on a complete explanation of the way laws work in the Vatican City, but I have found a "good reference". In a nutshell,
The VCS came into existence as a sovereign nation in 1929 with the signing of the Lateran Treaty between the Holy See and the Kingdom of Italy. The signing effectively ended the “Roman Question,” the decades old tension between the Catholic Church and the nation of Italy. Prior to the treaty, the relationship between the Church and the country was governed by the Law of Papal Guarantees, an Italian law that allowed the Pope a certain amount of autonomy within the borders of Italy.
The Lateran Treaty consisted of three separate documents spread over twenty-seven articles and four annexes: an agreement acknowledging the Vatican as an independent state, also known as the Treaty of Conciliation; a concordat on church state relations between the city state and Italy; and a financial convention liquidating the financial claims of the Holy See against Italy. In signing the treaty, Italy ceded 108.7 acres of Rome to the Holy See, thus creating the world’s smallest sovereign nation. At the signing, Pope Pius XI was represented by Cardinal Pietro Gasparri, papal secretary of state, while King Emanuel III was represented by Benito Mussolini, prime minister of Italy. The Lateran Treaty was incorporated into the Italian Constitution sixteen years later in 1947.
. . .
On the same day that the Lateran Treaty was signed, the VCS adopted a constitution in the form of six constitutional (or fundamental) laws: Fundamental Law of the City of the Vatican; Law of the Source of Laws; Law on the Rights of Citizenship and Sojourn; Law on Administrative Organization; Law on Economic, Commercial, and Professional Organization; and Law of Public Security. Under the second of these laws, the sources of VCS law were comprised of the Codex Iuris Canonici (Canon Law Code), and “[t]he laws promulgated for the City of the Vatican by the Sovereign Pontiff or by any other authority delegated by him, as well as the regulations lawfully issued by the competent authority.” Article 3 of this law also allowed for the use of Italian law as well as provincial and municipal Roman law when they did not conflict with canon law, the rules of the Lateran Treaty (and, later, the 1984 Concordat), or divine law.
The role of the Pope
The VCS is a unique entity in that the state’s monarch is also the spiritual leader of the Roman Catholic Church throughout the world. As the elected absolute temporal monarch of the state, the Pope has full legislative, executive, and judicial authority over the jurisdiction. The Pope delegates most of this authority to a variety of organs within the Vatican City, all of whose members may be appointed or removed at the discretion of the Pope. The powers and duties of these various organs are described in detail later in this article.
In other words, everybody answers to the Pope. He may not oversee this particular case, but he theoretically could. More specifically, though,
The judicial system of the VCS is organized as follows: a sole judge (Giudice Unico) presiding over a court of limited jurisdiction; a three-judge Tribunal (Tribunale); a four-member Court of Appeals (Corte d’Appello); and, finally, the Supreme Court of Appeals (Corte di Cassazione). It is important to distinguish these judicial organs from those of the Roman Curia, which is the administrative arm of the Holy See. Cases from temporal VCS courts are not generally reported, but a listing of the types of cases tried before each of the courts is published in L’Attivita della Santa Sede, the annual yearbook.