How to tweak Ubuntu running on an old PC?

380MB RAM is pretty low, but should be fine running a lightweight OS.

If you want to stick to an Ubuntu based distro, I recommend Lubuntu. It uses the LXQt desktop environment. Watch out when installing heavy packages; try to find an alternative. Firefox is known to take a lot of memory, if the browser provided with Lubuntu suffices, stick to that one.

I'm using Lubuntu as portable distro on my USB flash drive, and it runs pretty smoothly. Xubuntu (Xfce DE) is a great option as well, but it’s not as light as Lubuntu.

Try Lubuntu. It has LXDE, which requires less RAM as all the other DEs. Check this one: https://wiki.ubuntu.com/Lubuntu#System%20requirements