I don't know if the Mongols would encounter Christianity without their forays into Persia and Asia Minor. They may perhaps pick it up eventually through trade, but I feel they would remain true to either their own polytheism or convert to other far eastern Religions like Buddhism or Daoism. I think Confucianism would be too traditional for the Mongols, and they would try and eradicate it.
A minority of Mongols were Christian already during Genghis Khan's time, so there is no doubt that the Mongols would be exposed to it. Nestorian Christianity did not die out in China until around this time anyways, so vestiges would remain.
Confucianism, while being very traditional, would be too important to the Mongols to get rid of: the scholarly tradition produced most (though I think technically not all) of the bureaucrats, which the Mongols would be dependent on for ruling their vast domain.