I have not use a LM358 so I cant say what the advantages are.? I just Used what I had on-hand.? A quick peek at the data sheet and it seems like it should work as a replacement.? I think the key feature you want is that the op amp output can swing to ground.
With the circuit you linked, there is rectification/smoothing in the Op Amp output.? I think this should should work for the SMeter, but I think the FFT/CW Decode features will not work properly as it expects the full signal (preferably centered at 2.5V)
Other than that, be sure that the signal into the Arduino doesn't exceed 5 volts.
If you get the LM358 to work, please let me know so I can update the page with the alternative.
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Dennis WC8C