Can neurons take oxygen from air?
Short answer
Neurons, and in fact all cells in the body, utilize water-dissolved oxygen to fuel their energy metabolism.

Background
The brain, and tissues in general, obtain their oxygen through passive diffusion (Fig. 1). Water, carbon dioxide, and oxygen are among the few simple molecules that can cross the cell membrane by passive diffusion,  or osmosis (source: Estrella Mountain Community College). 

The body uses hemoglobin to transport oxygen (Fig. 1), because passive diffusion would take too long, especially in oxygen-hungry warm-blooded animals like humans.   


Gaseous exchange in fluid medium is mediated by hemoglobin. source: Kokyo