You really need separate, linear, regulator for the mic preamp - the noise will be digital hash from the Arduino most
likely, nothing to do with the audio amp. Switchmode supplies or sharing digital and analog power rails are
two very good ways to get audible noise in such a preamp - microphone amps make microvolt level
signals audible.
The main difference between the Uno and Pro Mini that's likely to be relevant is the lack of large electrolytic
capacitors on the tiny Pro Mini board. That would account for 10dB of difference in noise nicely.