Intro
The necessary driver for wired networking is available in kernel 6.13.2-061302-generic
I didn't check any earlier kernels.
Detail
USB tethering
To get a temporary network connection, @Jeremy31's suggestion in the question comments of USB tethering worked very easily from my Samsung Android phone. This allowed downloading and updating.
Connect the phone to the PC using a USB cable.
In Android: Settings > Connections > Mobile Hotspot and Tethering > USB tethering ON
Adjust GRUB
You will probably want GRUB to show kernel options (Advanced Options for Ubuntu) in case the new kernel causes problems with your system:
The answer to How to get the GRUB menu to show up at boot-time in Ubuntu 24.04 has the detail: https://askubuntu.com/a/1521908
Install newer Kernel
The program 'mainline' makes this easy:
It's installation and use is described under 'Method 3: Manually Update the Kernel (Advanced Procedure)' in How to Update Linux Kernel In Ubuntu
sudo add-apt-repository ppa:cappelikan/ppa
sudo apt update
sudo apt install mainline
For me, 'mainline' came up as a terminal application (non-GUI).
mainline help shows the options.
sudo mainline list shows the Kernel versions available for install.
sudo mainline install-latest installs the latest version (in my case 6.13.2).
Then simply re-boot. The new, most recent, kernel is chosen by default. But 'Advanced Options for Ubuntu' can be selected so as to choose another kernel if the new one causes problems.
Almost certainly 'Secure Boot' in the BIOS will complain about the unsigned kernel. You'll have to disable secure boot until the official kernel version gets past the one you just installed.