The map to the pci.id is in modinfo.

$ modinfo e1000e | grep 1503

alias:          pci:v00008086d00001503sv*sd*bc*sc*i*

To install the updated e1000e, you will need to download the file to your desktop, extract it and then, from a terminal:

sudo apt-get install build-essential linux-headers-generic

cd Desktop/e1000e-2.3.2/src

make

sudo make install

sudo modprobe -r e1000e && sudo modprobe e1000e

Each time a new kernel, also known as linux-image is installed, recompile:

cd Desktop/e1000e-2.3.2

make clean

make

sudo make install

sudo modprobe -r e1000e && sudo modprobe e1000e

Obviously, to install all this, a temporary wireless connection is strongly suggested.