The biblical account of the flood of Noah's day says that the entire earth was covered with water.
At face value, that would mean the highest mountains would be covered by water - around 29,000 ft.
Would that result in the breathable atmosphere moving upward allowing Noah and his family to breathe without aid?

Some denominations claim that the highest mountains of the day were more likley to be in the range of 3,000 ft and the water pressure from the flood waters pushed up the peaks we have today.
Would that cause the mightiest of tsunami, wreaking havoc with the ark?
there are of course more related questions, but suffice for now...

Since you posted this question in a science forum, and not a religion forum, we should restrict this discussion to scientific issues, and not get bogged down in issues of faith.

Therefore let me start by saying that there is no evidence that a flood that covered the entire earth (as we know it today) to a depth that covered all mountains all at once ever occurred. There simply isn't enough water to do that. And it is absolutely certain that no flood of such magnitude has occurred in the past 10000 years or so.

But if the water level did rise to 29000 feet, then to answer your question - yes, the new "sea level" air pressure would occur at the surface of the ocean. This assumes that the density of air is not reduced through removal of all that water vapor which presumably precipitated out as rain, but since that's impossible anyway we won't bother going there.

There is no evidence that the average height if today's mountains is higher (or shorter) than mountains of the past - the process of plate tectonics, volcanic activity, and erosion results in a constant cycling of mountain ranges growing in some parts of the earth (like in the Himalayas today) while others are eroding away (the Apalachian mountains, for example). But it seems that if there ever was a cataclysmic event that caused mountain ranges all over the earth to rise all at once then clearly the earthquakes, tsunamis, environmental havoc, etc that would have resulted would be indescribable.