Is there a preferable way to amplify the signal from an electret microphone?
To get a gain of 400 with a Common Emitter (bipolar transistor) circuit you will probably need 2 stages and a dozen or so passive components. This is what I used in the last device I built that amplified and peak-detected a microphone input. But getting the component values right was a pain, and wiring it was no fun either.

An op amp can do the same job with fewer components, and is easier to 'tune'. But you need to choose one that works on the voltage you have available. If the power supply is 5V then to get 4Vpp you will probably need an op amp with 'rail-to-rail' output. 

Other op amp specs depend on the circuit used and the required performance. If it is operated with a DC gain of 400 then you want a low input offset voltage (&lt;1mV for 0.4V output offset), and if 'without a DC offset' means only amplifying the positive half of the waveform then the inputs will need to work at or slightly below ground. Gain Bandwidth Product should be high enough to amplify the highest frequency you are interested in (eg. at least 4MHz for 52dB at 10kHz). A dual op amp would give you an extra one for the peak detector. The MCP6022 is an example that may be suitable.