Can we train our neurons? Like the experiment where human vision adapted to upside down image, could our brains somehow adapt to understanding 4D data from VR headset?

I'm sure some form of training is possible where you get a better understanding of a 4D universe with some limited inference abilities, but with a bad analogy, this would all be "software emulated" with no hardware acceleration - we only have the latter for 3D and we can't update it without a hardware change.

Yes, I believe it's possible to train our brains and learn to perceive better in higher dimensions. There's a great description in the science-fiction book Neverness, where pilots meld their minds with the spaceship computer to visualize and navigate hyperspace.