You'll find a list of A1200/A600 compatible PCMCIA network cards here -
http://www.g-mb.de/pcmcia_e.htmlThe list isn't complete, so even if your card isn't listed, there's still a chance it could work. Drivers are available on Aminet, and there are links to the appropriate ones on that page (3c589, CNet etc).
As for software, AmiTCP will be the most memory efficient to run, although it harder to set up, but not impossible. Miami might run too, but loading MUI takes a fair amount of RAM. The same thing goes for AmIRC, so you'd be better of using Grapevine for IRC. ALynx should work for browsing, but it's only text based and very old. AmiMSN is command-line based so you could use a Shell for MSN if you wanted. I'm sure there are some email, news, FTP and Terminal solutions that should work in 1MB too.
To be honest, you'd be better off upgrading your A600, and taking advantage of more modern programs like AmIRC/WookieChat, IBrowse, SabreMSN, YAM and others.
You can buy a Kickstart ROM Switcher and install a 3.1 ROM on it, which is essential for some newer software. They're available here -
http://www.amigamaniac.comYou'll also gain a lot from installing an extra 8MB RAM. You can get an 8MB RAM card for the A600 from here -
http://amigasc.nm.ru/english/buy.htmlBest of luck getting your A600 online!