Yes. Absolutely higher as it falls, not lower.
Got it corrected before the edit window ran out.
I’m considering the clarified question:
I was considering it a question about the pressure changes on the portion facing down vs those upwards.
My understanding of the physics of drafting was what I was considering. Not sure why that is hard to understand?
@TriPolar I guess then my understanding of the physics of drafting has been wrong?
I understood that lower pressure forms behind the lead cyclist dragging on them. A cyclist behind them benefits by being in that lower pressure zone and to a lesser degree benefits the lead rider by reducing the the drag on them. Someone in the middle of a larger peloton gains the most benefit.
That effect, however I may be misunderstanding it, is big enough for me to be able to notice even at the speeds I was able to hit.
It will be lower than the ram air pressure on the lead cyclist. Not lower than the ambient air pressure. An ideal aerodynamic shape could possibly create a non-turbulent slightly lower pressure zone of air behind it. However, the air flow relative to the sides of an ordinary bucket won’t reattach smoothly over the top of the bucket so the result will be a turbulent zone.
Ah. So drag pressure, the higher pressure on the leading surface (ram air pressure) than in the area behind the object with those turbulent eddies, is created by the increase on the lead not any decrease from behind? It is not lower pressure to atmospheric only relative to the forward pressure zone?
Sure. As a kid on a bicycle cycling on a city street, when I was going upwind it was very helpful to get behind a city bus and be shielded from the wind!
But in the context of this question, we need to ask how important that dynamic pressure effect is to a falling bucket of water. The answer is, hardly at all. Atmospheric pressure at sea level is just a bit less than 15 PSI. The dynamic pressure load from traveling, say, at 30 mph at sea level, as you or I might have been at top speed on our bicycles, is around 0.016 PSI. The negative dynamic pressure would be much less than that. So, IOW, the “draft” effect you’re referencing is completely negligible relative to the static pressure of the atmosphere.
This is the part of the answer that I find so unsatisfying. It reads hand wavy to me. Why that scale of impact required in order to answer what the impact is? Is it a nonzero number? Does it reduce relative to atmospheric pressure at all?
Or IOW do you agree that drag pressure is all the increase of pressure forward and the relative low pressure behind is low only relative to the forward zone not to atmospheric?
To quote Sir Horace Lamb:
Except at low speeds the flow regime behind the bucket will be turbulent. Making definite statements about the pressure at the surface of the bucket is likely futile.
The bucket will begin its fall in free fall. So the water in the bucket will no longer be retained by gravity. Air resistance will stop the bucket staying in perfect free fall and as the bucket gets closer to its terminal velocity gravity will return as the dominant influence on the water.
Somewhere in the middle it is possible that the lower retention of the water, and reasonably powerful turbulent flow of air might whip the water surface up and some water may be lost.
Other than doing the experiment I would be inclined to answer with “it depends”.
Actually measuring a change in static pressure is remarkable. Cheap pressure transducers can register a change in pressure of a fraction of a metre change in altitude. If you have an iPhone, download Physics Toolbox, which is a free app that gets you access to the raw data from the phone’s various sensors. Just moving the phone up and down a short distance will register easily detected pressure changes.
But the effect on the water of pressure changes will be negligible. Maybe, just maybe, if you filled the bucket with water right at its boiling point, a slight reduction in pressure might see the water briefly boil. That would be a cool trick. But I suspect it would take a lot of tweaking to make work.
This is the sort of thing for a You Tube clip. Attach a GoPro to the handle and drop the bucket from a significant height.
Be sure to attach aerodynamic fins to the bucket or your experiment will be destroyed when the bucket goes sideways for any reason or no reason.
Right, while @Francis_Vaughan is quite correct, some of these lesser practical considerations obfuscate the basic question. The OP seemed to be conflating the effects of atmospheric pressure with the effects of gravity. To see the relevant major phenomena in the proper perspective, one needs to make some simplifying assumptions to eliminate the unimportant minor ones.
So we assume that the bucket is dropped from only a moderate height so that there’s no significant difference in atmospheric pressure between top and bottom, which also prevents the bucket from reaching speeds where aerodynamic forces and turbulence become significant. We assume the bucket somehow remains upright on the way down. We can then state that in this simplified model there is absolutely no difference in atmospheric pressure on the surface of the water whether the bucket is stationary or falling.
So to clarify my misunderstanding:
The region behind a lead bicycle or the underside of an airplane wing is only low pressure relative to the opposite side, not to the atmospheric pressure?
This is different than how I had understood explanations in the past so I do want to be sure I’ve got it right.
A lead bicycle or falling bucket are very different than an airplane wing. An airplane wing could be a neutral faired shape. If it’s an ideal shape consistent static and dynamic pressure can be measured in front, on the sides and behind the wing as it air flows around it. A bucket is nothing like that, air pressure will increase measurably below the falling bucket, probably be entirely turbulent as it passes by the sides of the bucket, and definitely become entirely turbulent as the air flow rejoins over the top of the bucket. The turbulent air has no measurable pressure at any specific point and changes over time so the volume of turbulent air is assumed to be at the same pressure as the ambient air around it. It’s that turbulent air that reduces drag on a bicycle drafting behind a lead bicycle. For some reason that I don’t know how to explain objects experience less drag moving through turbulent air. That’s what drafting does.
I did not mention that very close drafting behind a leading object can result in the accelerated air flow passing the sides of the lead object continuing past the trailing object eliminating the ram air resistance on the trailing object as if it were attached to the leading object.
Maybe that it’s lower pressure? 
I’m dealing with the popular media explanation I’ve heard forever, like this one:
And even wiki
And it makes sense. Ram pressure occurs because more particles and compressed. There is a paucity of particles in the wake, which creates less pressure, which why it fills in in that turbulent manner.
In the case I mentioned of two objects moving, one closely behind the other you can get that low pressure zone effect. But it doesn’t apply to a falling bucket, there’s no drafting going on.
So?
Drafting works because of the low pressure zone; it doesn’t create the low pressure zone. It helps the lead vehicle by making that low pressure zone less low, reducing drag.
The bucket falling compresses air forward and leaves a relative void behind it which gets filled in by air sucked in from relatively higher pressure into that lower pressure zone in a turbulent manner.
Hard to decipher what you are saying here but why do you think a low pressure zone behind a moving object is an advantage to a trailing object? The zone is moving at the speed of the leading object so it wouldn’t matter if it was a high pressure zone or a total vacuum because the trailing object is moving at the same speed as that zone. Close drafting works by eliminating the direct static pressure on the trailing object.
Because pressure is force per unit area? Less pressure is less resistive force to overcome. This part is basic enough for even me to understand.
Pressure drag is the net force created by the differential pressure, relatively higher pressure (force per unit area) in front compared lower pressure behind.
Having the pressure in front of an object being a lower pressure zone is of course an advantage and is how drafting works for the trailing vehicle.
The trailing vehicle for its part pushes air forward slightly increasing the pressure on lead vehicle (making the low pressure a bit less low), reducing drag on the lead vehicle.
I think that the wake behind the trailing vehicle is smaller as a result also, therefore not as low pressure as for a single vehicle.
Atmospheric pressure for its part is a different value at different heights, exact numbers here. At sea level it is 1.03 kg/sqcm and at 1000m it is 0.997, a drop of roughly 3%.
Answering the OP question with that figure alone seems … trite. I doubt the OP was only asking about whether or not air pressure changes with elevation.
The question is only interesting because of the aerodynamic pressure considerations.
Oh, I was. Well, that and the effect of gravity.
In that case - nevermind.
See the table I linked to for exact figures.