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.