If somehow Mars suddenly had our exact atmosphere, could there be lots of life in a few million years? Would the temperature rise quickly? Would the ice caps melt? Could the life be as advanced as modern humans?
Atmosphere is not enough - it would also need water.
And Mars already had both atmosphere and water: Mars would lose both new atmosphere and new water for exactly same reasons it did lost them before, in a similar timeframe (around a billion years or so) or even sooner (because now its core is less active, creating weaker magnetic protection). So if any primitive life evolved (and it would be harder on Mars, because its dead crust is much thicker and the planet is geologically almost dead) any new life would die out soon as water and atmosphere are stripped by solar wind.
So the answer is: No. Life would take many millions of years to evolve, and conditions would revert to the existing conditions pretty soon (in a geological sense).
Also as @jamesqf correctly noted, when life emerged on Earth, atmosphere was very different from present one, with no oxygen.
It is not obvious if life can emerge in any atmosphere with free oxygen like the terran one, because oxygen was poison for first primitive life, and current terran life still needs to protect its molecules from oxidation damage (by handling oxygen very carefully).
If you're asking whether life could arise on a Mars that suddenly became Earthlike, you have to remember that Earth life did not arise on an Earth that was at all like our present conditions. The atmosphere was probably mostly methane (though there are other theories), with little free oxygen. See e.g. the "Great Oxidation Event".
Short answer: No.
Long answer: We have two scenarios to consider. Four billion years ago Mars had Earth-like conditions and water in its liquid form, but we are still unsure if there was life, even in the form of bacteria. And even if it was, it is highly unlikely that it survived the last four billon years of very harsh conditions. So, scenario one, given that only the Mars' atmosphere changes to match Earth, but there is no life on Mars at all, a few million years won't change anything. You would probably need another billion for life to emerge via abiogenesis or for a viable "panspermia accident" to occur (ie. a meteor containing bacterias or other form of microbiological life hitting the surface in a way that doesn't destroy them and in a place which makes it possible for them to survive and develop).
Also note that even if you want Martian polar ice caps to melt, you would need atmosphere which would produce a massive greenhouse effect - denser and more richh in CO2 than Earth's atmosphere.
But, scenario two, let's say some extremophilic strains of microorganisms survived those billions of years or - more probable - our spacecrafts carried some on them. If the atmosphere changed suddenly to match ours, such microorganisms could reproduce and in a few million years they might evolve in ways very different from their Terran ancestors. Still, it would be just extremophilic microorganisms, living on a cold, desert planet.
I would point out, for arguments sake that:
- We do not know how life started on Earth.
- We do not know that there is no life on Mars.
As such, not only could it be conceivable that within a few million years with an Earth atmosphere life could originate on Mars, it's also entirely conceivable that it'd never happen or that it's already happening with Mars's existing atmosphere.