How to get and install kernel from kernel.org
You may grab the latest kernel source from Linus at kernel.org and compile your own kernel.
It is easier to just make a package for your kernel image.
Install fakeroot and kernel-package. Also, install linux-tree.
Linux-tree is the linux source, packaged by Ubuntu. It is a better starting point because it contains a lot of patches, bugfixes and configuration you need for ubuntu. You can make menuconfig and remove the options you do not need.
cd to the top of the linux tree and do
fakeroot make-kpkg --revision=1 --append-to-version=MyCustomKernel --initrd kernel_image kernel_headers
That will compile and build a kernel-image package that you can install with dpkg -i.
sudo dpkg -i ../kernel-image-188.8.131.52MyCustomKernel_1
It is easier to install and remove that way. It also guarantees that your system is not left unbootable (by not screwing up your BootLoader and leaving you other options).
How to get and install new 32-bit Ubuntu kernel from http://kernel.ubuntu.com/~kernel-ppa/mainline/
Run the following Terminal commands to install a new 32-bit Ubuntu kernel from http://kernel.ubuntu.com/~kernel-ppa/mainline/
sudo apt-get update
sudo apt-get install python-bs4
rm -rf medigeek-kmp*
wget --no-check-certificate https://github.com/medigeek/kmp-downloader/tarball/master -O kmpd.tar.gz
tar xzf kmpd.tar.gz
python kmpd.py -d
The script at https://github.com/medigeek/kmp-downloader/ does not seem to allow choosing amd64 (64-bit) architecture. It only allows installing i386 (32-bit) architecture. This seems like a bug to me. Just press <ENTER> instead of a number if you get stuck on a certain question in the python script.