How did Marc Antony and Cleopatra influenced the culture of their nations?

Neither one of them really influenced the culture of their nations. Antony was an ordinary Roman politician who got his position due to Julius Caesar. He followed the set procedure of a Roman aristocrat by studying in Greece, joining the army and entering politics. His name would just be another on the roll of Roman consuls if Caesar had not been assassinated and Antony, as Caesar's co-consul, thrown into the historical spotlight. His fortunes and misfortunes after Caesar's death did not cause him to influence Roman culture in any way.

Cleopatra, too, did not really influence Egyptian culture. Although an ethnic Greek, she was Egyptian culturally. Her family had ruled for over three hundred years and she was totally emerged in Egypt, She spoke the language, followed its religion, and ruled according the the Egyptian method of nones. She made no changes nor did she introduce any new customs.