*DONT* use a LM358 - without an output bias resistor to the negative rail you will get highly objectionable crossover distortion.
1. I once listened to some Burzum through a "CMoy" amp with LM358 (directly driving headphones) and it sounded perfectly fine, so objectionable my ass
2. It's only a problem when the opamp is loaded enough, hard to tell if this circuit is affected without knowing component values.
3. There are better reasons to avoid LM358 if you are a Hi-Fi snob, such as noise and limited high frequency performance.
I was hoping that someone has build that composite amp.
You may have more luck with that on the DIYAudio forum than here.
If it would help I can take a very close picture.
Definitely take a close look at the chips you have, there should either be markings or at least some sign of them having been removed (scratches, maybe black paint). Sometimes markings are obscured by flux residue or other junk, so try cleaning the chips with alcohol or acetone.
Lastly, I had a look at the schematic and can tell this much:
U1 is a simple preamp, fairly noncritical for circuit operation. Any dual opamp will work here, the only difference being performance. The author says it's "high performance" so most likely at least NE5532 or LM4562, or one of the fashionable these days OPA16xx series from TI. Given a DMM, breadboard and some resistors, I could identify NE5532 or LM4562, but the TI parts could be more work because there is many of them.
U2 is part of a feedback loop around the output stage, so its high frequency performance may affect stability of this whole composite thing. This chip may be more sensitive to substitution. The pinout implies it's a single, so maybe NE5534 or one of the OPA16x1. National also had some LME49xxx single audio opamps, but I think they have all been obsoleted by TI.
For U2, start with checking for any resistance between pins 1,5,8 and either of the supply rails. And see what's the value of C71 and R8, because it could possibly be a JFET input opamp. Maybe OPA134, OPA627 (crazy but why not), OPA1641, or something from AD/LT.