Your grandma is faster than me. I have an Acer M1100 and I'm still using it.
If the original Socket is an AM2 and the replacement processor is an AM2 as well there should be no problem but sometimes the BIOS version must be updated to receive a better performing processor that is not the same as what it was intended to do. The BIOS could be complicated as I do not expect ODM like Acer will ever release a BIOS update. A 2.1Ghz dual core seems acceptable still by today's standards.
I suggest that you try to install the XFX Radeon HD 5550 first to see whether there is any improvement. If the problem has not improved then you can consider changing the processor as well. But from what I can tell a GC in that PC should solve the problems already. A clean format of the PC to Windows 7 should speed things up a little as well.