conceptually yes, it amounts to a similar result as you say. The gotcha in LM358 which is the opamp indicated in the pic and some other similar workhorse single supply opamps is that they don't swing rail to rail... at least in single supply operation... it swings between GND and V+ - 1.5V, so powered with an Arduino, the output can swing GND to 3.5V... so you need to bias the audio input signal with resistor voltage divider to around 3.5/2 ~ 1.7V.
So the usage is -
Audio signal line wire 1 -> capacitive coupling -> voltage divider -> amplifier circuit input-> amplifier output -> capacitive couple for speakers or direct connection to Arduino analog pin and do analogRead() (you need to subtract bias voltage value)
Audio signal line wire 2 - connect with GND