I was just wondering, if I took some infinitesimal section of a fluid and constructed a free body diagram on it, would it experience a buoyant force from the rest of the fluid it is part of to cancel out its weight? And how does air pressure play a role in this situation?
To the first question: yes, formally. Keep in mind that small currents and diffusion means that the fluid will still mix, regardless of whether it's gas or liquid.
To the second question: the role air pressure plays is to set the pressure at the surface of the liquid. You should think of the buoyant force as being less about pressure, though, and more about the rate of change in pressure. The reason you experience a noticeable buoyant force underwater is because the pressure is changing with the depth linearly, producing a constant force, and your density is about the same as water's. Out in the air you also experience a buoyant force, but it's much smaller because the rate at which the pressure drops is much slower because air is about one thousand times less dense than water.
Your infinitesimal section of fluid is in static equilibrium and so the net force on it must be zero.
The section has a weight and therefore must have an equal in magnitude but opposite in direction force on it which is the upthrust produced by the rest of the fluid.
Replace your section with an object which has less weight (less dense) than your section and the net force will be upwards - air bubble in water, more weight (more dense) and the net force will be downwards - sand grain in water.
Air pressure has no effect as the upthrust depends on pressure differences unless your fluid is compressible.
A higher air pressure would compress the fluid making it more dense and so increase the upthrust.
For a liquid which is virtually incompressible this effect would be very small but for air below air (our atmosphere) the effect is significant.