No... Nor does the author say that.
The pre-oxygen atmosphere of the Earth mostly consisted of nitrogen, ammonia, carbon dioxide, and water vapor.
Later after plants arose with photosynthesis, oxygen appeared in the atmosphere.
Yes, around one billion years ago -- which is around a billion years after the appearance of the first eukaryote. (Oxygen was being produced long before that, but it was rapidly reacting with various compounds, including iron, until finally enough was produced to remain free in the atmosphere in significant amounts.)
The author wants us to believe that one or two methane utilizing bacteria combined to produce a eukaryotic cell that is oxygen based.
No he doesn't. You're wrong on the methane, and you're wrong in claiming that the author in any way asserts that the first eukaryotic cell had an oxygen-based metabolism. Clearly, oxygen consumption was developed long after the first eukaryote.
I don't think so.
And neither does the author.