No, the LM358 will not work for your application as described. But not due to the input offset voltage.
The input offset voltage is the voltage that must be applied across the input terminals for the output to be forced to 0V (or whatever the output is referred to and what you're choosing as ground). That might not be the most helpful description though, so I like to actually see the input offset voltage in the circuit.
The input offset voltage can be thought of as a voltage in series with one (but not both) of the inputs. It doesn't matter which one, and the offset voltage can be of either polarity, and fall anywhere in the range specified in the data sheet. It is a source of DC error, as whatever you amplify will be the voltage you're interested in ± that offset voltage which is in series with one of the inputs. So even if you are amplifying 500µV, and your offset is worst case around 2mV, and you want a gain of 100, then 0V input will result in 0.2V on the output. 500µV on the input will result in 0.25V on the output. That is certainly amplifying the signal, but there is just that error superimposed on it.
So the input offset voltage is simply that, a DC voltage offset that is always present, and will be amplified along with whatever else is at the input. If you care about measuring a specific DV voltage level and need that amplified, then the offset does indeed present a very real problem. However, when amplifying AC signals, it is not that much of an issue. It will reduce your dynamic range and maximum gain, as 2mV of DC voltage will be 2V on the output with a gain of 1000, so that is 2V less range your output can swing, so clipping can occur.
The bad news is this offset voltage is going to be fairly random even between devices that were made on the same wafer. One might be 2mV, another 1mV, who knows. Worse, the offset drifts with time and temperature.
The good news is this contributes to DC error. You're amplifying an AC signal. It will be AC coupled. Remember, the DC offset is just that - a DC voltage that the AC signal will be superimposed on. Blocking the DC component with a capacitor will leave you with the AC signal.
So, with only that in mind, the LM358 would work perfectly. Only, it won't. It's too slow. Sorry. It has a gain bandwidth product of 0.7MHz. That means it's gain falls to 1 at that frequency. Since your gain is 100, it will only be able to amplify signals that are 7000Hz (700,000Hz / 100) or less. Signals above that will see substantially less than 100 times the gain, which means you'll have significant frequency-dependent distortion on the output.
Now, if you can settle for a bit less gain, like 80 instead of 100, then you can retain full amplification over your desired frequency range. In that case, the LM358 will work fine. Well, "fine". It has a lot of cross over distortion, but I doubt that will be a significant degrader of audio fidelity when your input is an electret microphone. That is to say, the LM358 is a fairly crappy audio amplifier, but it works great if your audio is even crappier, which it definitely is going to be coming from a narrow range electric microphone.
Reduce your gain to 80 and don't expect CD quality audio or anything, and then the LM358 should work ok.
Besides gain bandwidth product (which is probably the most important parameter for amplifying AC signals, in that if it is too low, then you simply cannot use that op amp and will need to find a faster one. But as long as you are not asking it to surpass this, it stops being particularly important. It's an all or nothing parameter), things like CMRR (which is how well the op amp will reject signals that are common to both inputs - in other words, all the crap that ISN'T the signal), input voltage noise, and voltage gain into a load are probably pretty important.
Input voltage noise is amplified by your gain just like your signal and offset, so if your input noise is 50µV, that is much more like the hard limit you are imagining offset voltage to be. Any signal that is 50µV or less will just be drowned out in the noise, and there will be random variations of 50µV and everywhere below constantly superimposed on the input voltage, so keep that in mind. I like to imagine noise, offset, all that stuff as voltage sources in series with an input, and indeed, they can be modeled that way. But I find it helps me from an intuitive understanding position too.
Voltage gain, usually measured in seemingly silly units like V/mV, is how much gain you can expect with a certain amount of loading on the output. Many op amps will begin to lose a lot of their amplification if you start demanding they drive too low an impedance load, so just check that to be sure.
Anyway, op amps are thankfully a bit easier to use and there are less critical parameters if you're using them in an AC amplifier application. As long as its not too slow, not too noisy, and the input can drive the input impedance and the opamp's output can drive whatever impedance is loading the output, you generally won't run into any surprises. Things get a lot nastier if you're using them for DC stuff.