Shoulder-roll (aka philly shell) is the defensive boxing technique that got Floyd Mayweather Jr a career record of 50 wins and 0 losses. But that is not what it means here.
Short answer: "shoulder-rolling" here refers to the Kingpin, Tony the tiger, rotating his shoulders outwards/backwards to attain a posture that shows he is confident, authoritarian, and intimidating. See details to understand why and how.
From what I understand, this is a comedic story/play (playing in the theaters) called "Married to the Mob". The play is about the life of Italian gangsters and their wives.
In the following sentence,
Appealing as they are, the two leads are readily upstaged by Miss Ruehl and, especially, by Mr. Stockwell.
the two leads performing on the stage are Michelle Pfeiffer (her stage name is Angela, she is the heroine, widow of the gangster known as Cucumber) and Matthew Modine (an FBI agent keeping an eye on Angela; Angela kissed Tony who killed Cucumber). The two leads are appealing to watch, but they are quickly removed to the far end of the stage, and attention is brought on to Miss Ruehl and Mr. Stockwell.
"Upstage" (verb) means "to take people's attention away from someone and make them listen to or look at you instead." The adverb/adj form of the word means "towards or at the part of a theatre stage that is furthest from the people watching the performance." (Cambridge)
Mr. Stockwell (his stage name is Tony, the head of the gangster family, the "kingpin") steals the limelight.
His shoulder-rolling caricature of this suave, foppish and thoroughly henpecked kingpin is the film's biggest treat.
Let's look at some definitions from Cambridge first:
"Caricature" means "drawing or written or spoken description of someone that usually makes them look silly by making part of their appearance or character more noticeable than it really is."
"A suave man is very polite, pleasant, and usually attractive, often in a way that is slightly false."
"Foppish" means "(of a man) extremely interested in [his] appearance and wearing decorative clothes"
"A henpecked man is controlled by and a little frightened of a woman, especially his wife."
There is something about Mr. Stockwell that is the film's biggest treat. He is the head of the gangster family: the kingpin. He puts in a lot of effort in maintaining his "kingpin" appearance and status; he puts on decorative clothing (which distinguishes him from his subordinates) and carries himself around like the mob boss he is.
He is also concerned about his physical posture. The kingpin must look like he is invincible; he must stand like Superman. This is where the phrase "shoulder-rolling caricature" comes in. He rotates his shoulders backwards, and puffs out his chest (this is the part of his appearance that is hence more noticeable). This makes him look confident, authoritarian, and intimidating. But some of that is just his public persona: a facade. This is hinted by the use of the word "suave" (note the use of "slightly false" in the meaning).
What makes all this funny or ironic is that he is a "thoroughly henpecked kingpin". He is controlled by his wife; he is a little afraid of her. Think of the puffed-chest and the rolled-back shoulders disappearing when his wife, which a spatula, starts yelling at him for leaving the bedroom dirty.