Who can name the comedic movie rhyme that first advises to drink from
a royal cup than a mortar, but then later advises the latter instead
of a container decorated with a mythical beast.

The rhyme you're looking for comes from one of my favorite comedies,
"The Court Jester," in which this memorable dialogue occurs:

Hawkins: I've got it! I've got it! The pellet with the poison's in the
vessel with the pestle; the chalice from the palace has the brew that
is true! Right?

Griselda: Right. But there's been a change: they broke the chalice
from the palace!

Hawkins: They *broke* the chalice from the palace? 

Griselda: And replaced it with a flagon. 

Hawkins: A flagon...? 

Griselda: With the figure of a dragon. 

Hawkins: Flagon with a dragon. 

Griselda: Right. 

Hawkins: But did you put the pellet with the poison in the vessel with the pestle? 

Griselda: No! The pellet with the poison's in the flagon with the
dragon! The vessel with the pestle has the brew that is true!

Hawkins: The pellet with the poison's in the flagon with the dragon;
the vessel with the pestle has the brew that is true.

Griselda: Just remember that. 

Internet Movie Database: Memorable Quotes from The Court Jester (1956)
http://www.imdb.com/title/tt0049096/quotes

I hope this is precisely what you need. If not, please request
clarification; I'll be glad to offer further assistance before you
rate my answer.

Best regards,
pinkfreud

Super

Many thanks for the five stars and the nice tip!

~pinkfreud