You only need to measure the amplitude of the input audio which would be no problem to the Arduino. The input from the ADC is 10 bits. If that would give sufficient range for your intended application then it should be easy enough to do.
Panici:
Anyone have the relevant math equations handy?
No, but keep in mind you have/need three results you are looking for:
Attack time
decay time
Logarithmic amplitude display value.
A true VU display is a logarithmic scale from 0 amplitude to typical +3db where 0db is some standard reference value as shown here: VU meter - Wikipedia
If you just display a linear scaled value from the analogRead() value you will find that you will be able to display only a very narrow range of volume and most likely you will be very disappointed with the effect.