Why couldn't Goku use the Nimbus to rescue the Ox King from the fire? At the end of Dragon Ball, when the Ox King was trapped, Goku flew into the fire on the Nimbus and had an extended conversation with the Ox King. Goku's previous demonstrations of strength also implied that he would have no problem lifting and holding on to the Ox King.

I did not read the manga, I only watched the anime. Is any specific reason given or implied why Goku could not have grabbed the Ox King at that time and carried him (either on the cloud or hanging onto him if Ox King could not fit or ride) out of the flames to buy himself and Chi Chi extra time to find a way to extinguish the flames?
It was most probably because Ox King did not have a pure heart, and so would not have been able to ride the Nimbus. Evidence for this is seen during the first meeting of Goku and Ox King, when rumors that Ox King murdered those who approached his castle for treasures (out of greed, of course) was confirmed by him when he admits his wrongdoings in front of Master Roshi.

Anyways, it is filler content and not part of the original manga, so no "official" explanation exists.