Your laptop has the HM70 Express chipset. This chipset is artificially limited to using only Pentiums and Celerons, and the G2020 is close to the fastest CPU you can put in it. The only faster compatible chip is the G2030 and the difference is so small I wouldn't bother changing it.
If you put a "Core" CPU into an HM70 Express chipset board, it will auto shut down after 30 minutes. This isn't due to heat or power or anything of that sort, it's just an artificial limitation and there's nothing you can do about it. Intel did this so they could sell the chipset cheaper to manufacturers trying to make inexpensive laptops, so those manufacturers wouldn't use a really cheap chipset with a high-end CPU. It's very much like how a Pentium is the same physical CPU as an i3 (or even mobile i5 or i7), just with parts of it disabled.
So no, it won't work, and there aren't any CPU upgrades for your laptop.