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The theoretical resolution values are 0.0125mm (12.5 microns) for the X and Y axis and 0.004mm (4 microns) for the Z axis. This is determined by the steps of the stepper motor and the mechanical setup, you can investigate this topic by fiddling with the Josef Prusa RepRap Calculator: http://prusaprinters.org/calculator/.
However, the real system is affected by many errors (backlash, slack between parts, deformation…) that make these theoretical values not very significative. The most relevant values are the empirical values obtained by using the printer. We have tested it up to 0.1 mm (100 microns), which is a value typically marketed as “high definition” in FFF (Fused Filament Fabrication) printers. The M Prime One works very good at 0.1mm. You could also try lower heights just by setting that value in the slicer configuration.
About the controller board, any board will be able to print at those resolutions.