"Old-world Jewish culture" in this context would refer to the Jewish ghettos and villages in Poland, where Isaac Bashevis Singer grew up. It is exemplified by strong traditions, tightly-knit communities, and strong faith and religious influences. In "Gimpel the Fool," Singer uses the classic fable structure to show how people use others for their own amusement.
The most obvious influence of old-world Jewish culture in the story is the local Rabbi, who Gimpel visits for advice on the many tricks played on him by the villagers.
I went to the rabbi to get some advice. He said, "It is written, better to be a fool all your days than for one hour to be evil. You are not a fool. They are the fools. For he who causes his neighbor to feel shame loses Paradise himself."
Gimpel visits the Rabbi twice more during the story, and each time is given advice, but the Rabbi also refuses to take Gimpel's word for the indignities he undergoes, especially at the hands of his wife Elka.
Another example comes in Gimpel's profession, that of bread-maker; in most villages, this was an honored profession, being of vital importance to economy, hunger, and religious rituals, but for Gimpel, it is simply a job in which he is mocked:
A student from the yeshiva came once to buy a roll, and he said, "You, Gimpel, while you stand here scraping with your baker's shovel the Messiah has come. The dead have arisen."
"What do you mean?" I said. "I heard no one blowing the ram's horn!"
He said, "Are you deaf?"
However, the job permeates his thinking, so when he is suspicious of the motives in matching him with Elka, he thinks, "No bread will ever be baked from this dough."
Gimpel's wedding is also a good example, as the entire village gets involved, setting up the ceremonies, cooking food, and donating money and gifts:
There was singing and dancing. An old granny danced opposite me, hugging a braided white chalah. The master of revels made a 'God 'a mercy' in memory of the bride's parents. The schoolboys threw burrs, as on Tishe b'Av fast day. There were a lot of gifts after the sermon: a noodle board, a kneading trough, a bucket, brooms, ladies, household articles galore. Then I took a look and saw two strapping young men carrying a crib.
(Quotes: Singer, "Gimpel the Fool," salvoblue.homestead.com)
Even in the act of deception, the villagers make the wedding a joyous occasion, and despite his misgivings, this is the only place in the story aside from the very end where Gimpel seems to have a good time.