I installed Debian 10 Buster on the Asus Eee PC 1201K, with only 1 Gb of RAM and 150 Gb of disk. Notice that the CPU requires an i386 architecture. The installation is quite slow, but it completes without problems.
I encountered a problem at the first boot after installation: the boot process failed with several ACPI errors and a nasty PCI: Fatal: No config space access function found. The boot finished into the initramfs prompt where I can check that no kernel modules were loaded and no devices were available, mainly the hard disk was missing. Loading all the kernel modules via modprobe do not solve anything, the kernel does not detect the hard disk.
I finally solved the problem by installing the linux-image-4.19.0-0.bpo.6-686 and removing the pae version: linux-image-4.19.0-8-686-pae.
Installing a new package on a non-booting installation is not trivial, here it is a step-by-step guide:
Now you can mount the Debian installed into the hard disk and do a chroot in it:
mount -t ext4 /dev/sda1 /mnt mount -o bind /dev /mnt/dev mount -o bind /dev/pts /mnt/dev/pts mount -o bind /proc /mnt/proc mount -o bind /sys /mnt/sys chroot /mnt /bin/bash
Inside the chroot environment, you can download and install the new kernel package and do whatever you need.