Power Arduino Pro Micro from MCP2025 Vreg
The data sheet for the MCP2025-500 says about the Vreg output: -


  Positive Supply Voltage Regulator Output pin. An on-chip Low Dropout
  Regulator (LDO) gives +5.0 or
  +3.3V at 70 mA regulated voltage on this pin.


If your Arduino draws less current than this then I would say, with caution yes but beware of current surges from the Arduino. If it draws more then no. Also note this: -


  The regulator requires an external output bypass capacitor for
  stability. See Figure 2-1 for correct capacity and ESR for stable
  operation.


You should check to see what current the arduino takes under all circumstances that you might use it.