It's no problem I used Ubuntu on it some 12 years ago because Ubuntu was one of the few/first supporting all drivers out of the box. (Was the first and only time I used Ubuntu, I was really surprised how well it ran).

1 GB is absolutely no problem, really like NO problem.

It did everything, catching and recording time delayed(buffered) dvbt video via USB radio stick, chrome with multiple tabs, having thunderbird open all the time, downsizing torrents, using the xmms player integrated in fvwm via shortcuts.
Normal compiling tasks via gcc etc are also absolutely in the range of this machine, you don't need to wait years for it to finish.

Just run fvwm or another window manager that's low on resources.
Rox filer.

The EeePC is so nice, looks like shit but really nice. Finally a Sub-notebook with full keyboard any pleb could afford.