Archer's disguise is a reference to the classic comic character Father Guido Sarducci. Pam refers to him as Father Guido Sar-Douchebag. It also looks very similar to his "Bob Belcher" alter-ego, but clad in a cassock instead of a shirt and greasy apron.
Cyril acts like he has a phobia of priests and nuns stemming from a past trauma he's uncomfortable discussing. He tells Cheryl he is faking so he won't have to go on any more missions with Archer, but was shown reacting fearfully to Lana in a nun outfit during a flashback to an S&M scene. However, Lana and Cyril are dating during this episode so Cyril could have been faking his phobia outside of work to make his story more convincing.
The entire episode is a nod to the Robert Ludlum book "The Road to Gandolfo"
On the plane, and later in a cafe in Rome, Woodhouse is shown scratching, which is used as an indication that he is withdrawing from heroin. Actually, heroin causes users to itch while using, not while withdrawing. This misconception is also seen in the episode, "Blood Test".
When sitting in the café, Lana asks Woodhouse if there's a particular amount of heroin he can take without getting all Trainspotty, a reference to Irvine Welsh's novel Trainspotting.
Cypher is an X-Men whose special ability is easily understanding languages. Gambit is also mentioned. He is another member of the X-Men who can throw things and cause them to explode, usually cards.
Archer sarcastically says to Pam "Nice job Oliver Cromwell". The English ruler was famous for killing many Catholics.
We can now add the Camorra to Archer's list of enemies, and the legendary incompetence of ISIS has spread further.
Archer reveals that Dicky Woodhouse is dead and he not only did not allow Woodhouse to attend his funeral, he has not even told him about it.
While on the trunk of the Swiss Guards' car, Archer says "Who am I William Safire?" Safire was an American author and speechwriter who wrote a New York Times Magazine column called On Language.
Lana calls Woodhouse "Junky Brewster", a reference to the TV show Punky Brewster.
When Pam asks the Pope, "Can I squeeze in a confession?" he responds in Italian, "Oho, ora dov'e il tuo Martin Luther?" which translates to "Oh, where is your Martin Luther now?" Pam then responds, "Fine, I'll buy a damn indulgence!" Martin Luther is the founder of Protestantism, and scandals over the selling of indulgences set the stage for the Protestant Reformation.
Archer lands on the hood of a Guardia Svizzera cruiser in a pose evoking "The Creation of Adam," by Michelangelo, although his left knee is raised instead of the right. Presumably, this alteration is to cover his junk for censors.
When Archer sees the pope sleeping, he mentions how he likes to spook Woodhouse by sitting by his bed wearing night-vision goggles. This prank was mentioned in the How to Archer book.