Can I run an Arduino Nano with 12V for this VU meter project?
No, the Nano takes a 5V power supply.  12V will kill it in an instant.  If you have a 12V supply you will need to use a voltage regulator or other voltage converter to get 5V to supply the Nano.
Keep in mind that the Nano's input/output pins are NOT 12V tolerant. So you will need to ensure that you keep all voltages that the pins are exposed to at 0-5V.
A circuit similar to this will get you 5V from a 12V source although there are many other possibilities.



simulate this circuit &ndash; Schematic created using CircuitLab