One of my favorite things about answering questions here is that when I find a question that I don't know the answer to, but I know where to look, I end up learning so many new things.

Because the answer to this one was both fascinating and somewhat hilarious.

The first encounter between a Kaifeng Jew and a Western Christian was actually not in Kaifeng, but in Beijing, in 1605. A Jew (though he called himself instead an "Israelite") named Ai Tian, a shidafu (scholarly official) from Kaifeng who came to Beijing in the hopes of passing exams necessary for a better government position, had read a popular Chinese book called Things I Have Heard, in which the author described the two Western men who had recently taken up residence in Beijing in terms of wonder. They were described as having banished demons from a house and owning a picture of the "Lord of Heaven" (from the description, obviously meant to be the Pope) in which his eyes follow you around the room, among other things. They were also described as monotheists who were insistent that they were not Muslims, and Ai, never having heard of Christians, came to the obvious conclusion- they must be Jews.

The idea that these two men were Jews was very exciting to Ai, though he wasn't personally particularly pious or Jewishly learned- the Jewish community of Kaifeng had been isolated for a few generations from non-Chinese Jews, ever since the end of the golden age of the Silk Road, and the prospect of reopening communications with other Jews from other parts of the world was one that he couldn't miss. So he went off in search of these Western Jews when he arrived in Beijing, and soon found their place of worship, which he took to be a synagogue.

In the meantime, these two men, actually Catholic priests named Matteo Ricci and Lazzaro Cattaneo, were, among other things, in search of the Christians whom Marco Polo had, hundreds of years earlier, claimed lived in large numbers in China. They had had a lead from Muslims to the existence of a community of the descendants of these Christians, and then one day, a man knocked on the door of their church who claimed to be a coreligionist. Ricci, who recorded the meeting, was ecstatic- apparently he had found one of these purported Christians!

Ricci invited Ai into the church, where there were several pictures at the altar- one a picture of Mary and baby Jesus, one a picture of St John the Baptist, and another a picture of Matthew, Mark, Luke and John. Ricci knelt in front of the pictures, and Ai looked curiously at them. His conclusion was that they, respectively, depicted the matriarch Rebecca, Jacob, Esau, and the four eldest sons of Jacob, and he knelt in front of them too out of politeness, though he told Ricci that in his religious community, they did not kneel. He complimented Ricci on the pictures, but told him that he was confused- why were there only depictions of four of Jacob's sons? What about the other eight?

Ricci was dumbfounded, but he had the knowledge to realize what was going on- that while he hadn't been lucky enough to encounter a Chinese Christian, he had instead found someone even more unusual, to the point of being hitherto unknown in the West- a Chinese Jew. Ricci, coming from Italy, of course knew what Jews were, but Ai did not have the same context, and so it was a lot harder to convince him that Ricci and Cattaneo were not Jews at all, but something new called Christians. As it turned out, Ai was able to confirm to a thrilled Ricci that the Chinese Christians he sought did exist, though they were known instead as "worshippers of the cross," known for making the sign of the cross over everything they ate and for marking their children's foreheads with ash in the shape of a cross, and had since converted to other religions. (Later, when Ricci tried to make contact with them, they refused, as they didn't want to jeopardize the assimilation into Chinese society which they had achieved.)

But Ricci could simply not make Ai understand that Christians were not Jews. To Ai, while they had many strange customs (such as eating pork) and believed, unlike the Jews, that the messiah had already come, they were just Jews in a different guise. Nor, it seemed, could the rabbi of the Kaifeng community understand; following his meeting with Ai, Ricci sent a letter to the rabbi, explaining Christian theology. The rabbi wrote back, saying that while he could not agree with Ricci's assertion that the messiah had already come, otherwise, Ricci seemed like a nice Jewish boy. In fact, said the rabbi, he was so impressed with Ricci's theological knowledge that he was willing, given his advanced age, to appoint Ricci as his successor to the rabbinate of Kaifeng (and effectively of China). There was only one condition- apparently, Ai had told the rabbi that Ricci ate pork, which he could not do if he were to become the rabbi of Kaifeng. Sadly, there is no record of Ricci's response to this generous offer.

The first Jesuit missionaries in Kaifeng didn't arrive until 1628, and it seems that by then the Jewish community had wised up to the differences between Judaism and Christianity; while the Jewish community is known to have had relationships with the various missionaries who served there, there isn't a single extant record of a Kaifeng Jew who converted to Christianity.

Sources:

Fontana, Matteo Ricci : A Jesuit in the Ming Court

Pollak, Paper and Laytner, "Detailed History of Kaifeng Jews"