Electrical – Photodiode circuit Op Amp and amplification failure

Without knowing your illumination levels, it's impossible to tell if that will do what you need. That said, the circuit ought to work, with a couple of changes. First, eliminate the resistor from pin 3 to ground and connect directly to ground. Next, increase the 15 pF capacitor to 100 pF. Your actual photodiode has much higher capacitance than the schematic, and this needs to be reflected in the feedback cap. Worse, it interacts with the + input, and gives a big transient spike. In the original, the diode capacitance was reasonably compensated by the ICs input pin capacitance, but that's no longer true with your new photodiode. The resistor was there originally to compensate for temperature changes in the input bias current.

It's entirely possible that you have way more light coming in than this setup will handle. If so, and your output is pinned high, reduce the value of the feedback resistor. If the signal is too small, increase the resistor value, although this may also require increasing the feedback cap, which will slow the response. I don't see this as a problem, though, since you're sampling at a very low rate.

The circuit on the right side makes use of an LM 108 and a capacitor on pin 8 ( due to the datasheet pin 8 of the op amp stands for Comp). Not having this pin on L358 what can I do for that?

The LM108 has pins for external compensation. As you've already figured out you will not find these pins on the LM358 because the LM358 is internally compensated (e.g. the capacitor is within the chip). So if you use the LM358 you don't need that capacitor.

Morever the arduino analog pin normally indicates a value btw 0-1023. Now using the op amp I get a maximum value of 767 ( this means approx. 3/4 of 1023). Is there any particular reason for that or am I doing any mistake?

The output of your amplifier does not generate a voltage that the arduino would interpret as full scale. You can increase the amplification by choosing a bigger resistor between the output and the negative input. If you want your amplification 25% higher just make the resistor 25% bigger.

Note that the voltage at the LM358 output can't go as high as the supply voltage. It will always have a maximum output voltage that is 1.5V to 2V below the supply. That means, if your arduino analog input would give you a full scale at - lets say - 5V, then you need to supply your OpAmp with 7V at least. Otherwise the output will clamp and never reach the full scale.

You'll also likely don't want to connect your OpAmp output directly to the analog input. Analog inputs have a maximum input voltage, and you should never exceed this. If you do current will flow through a protection diode. If the current is to large that could blow up your arduino. Adding a resistor between the OpAmp output and the analog input will limit the current without influencing the reading much. 2.2k is what I usually use as a rule of thumb value.