I had the 78 RPM record as a kid in the 50's, too. I probably still have it, in storage. It was sung by Arthur Godfrey. I think he got rid of the monkey getting drunk because it was for children. The version I recall is:

I went to the Animal Fair;
The birds and the beasts were there.
The big baboon, by the light of the moon,
Was combing his auburn hair.
You ought to have seen the monk,
He sat on the elephant's trunk.
The elephant sneezed, and fell on his knees,
And what became of the monk, the monk, the monk, the monk, the monk,
The monk, the monk, the monk,
And what became of the monk?

On the flip side was The Bull-dog.