Hi DiAno
Welcome to the forum.
The light side:
I would suggest trying a 32 bit version (X86) because of your limited RAM.
Try something relatively recent, like TC14, 15, or 16.
The dark side:
I have an Asus T100 netbook with an Atom Z3775.
It has 2 Gig RAM soldered in, not expandable.
It refused to boot a 32 bit kernel, only 64 bit booted.
I had to config and compile a new kernel.
A new hope:
I did eventually get most things working, including the touch screen.
I settled on running Core64 (32 bit apps on 64 bit kernel).
My kernel config changes were added to later TC versions.
If your Asus EeePC is anything like my T100, go to the search page:
https://forum.tinycorelinux.net/index.php?action=searchAnd set the Search for: field to asus t100
You'll see more information I've posted on this device.