To make it a little easier, since your comp only has 64Mb of RAM the changes of running a live CD or even a GUI is small. Your best bets would be to try Puppy (which you have) or possibly Arch Linux. You can attempt to install Slackware, but if Arch doesn't work then Slackware won't either.
The only other option is for you to install a distro that has no window manager and runs in Command Line Mode, which means that you can't use apps like firefox or anything Graphical User Interface based. But in CLI mode you can use links or lynx to access the internet in a text based browser.