This episode was in the middle production when Pope Benedict XVI announced his retirement, and aired two weeks after the selection of Pope Francis I.
This is the last episode to feature the voice of George Coe. He passed away in 2015. His character, Woodhouse, was voiced for 16 episodes by Tom Kane before Woodhouse was written out in Archer Dreamland: No Good Deed (2017).
To learn how to go undercover as a priest, Sterling watches Lucy: The Daughter of the Devil (2005), which featured a group of Vatican agents called the Special Fathers. H. Jon Benjamin provided the voice of Special Father #1 on that show as well as the voice of Satan.
Archer impersonates a priest at the Vatican. The mustache, glasses, and ensemble are a take-off on Don Novello's Saturday Night Live (1975) character, Father Guido Sarducci. On the plane, Pam alludes to it with "Father Guido Sardouchebag".
On the plane and later in a café 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.