Support for Avast 4.8 is likely to be withdrawn around the end of this year.
There is no point in attempting to update the program on your computer, just the database. The new program will not work in '98.
Not many AV's actually will work in W'98, I think Clam might be one (demand scanner only), if it does, maybe you could use Spyware Terminator which can optionally have Clam active, to use as a line of defense.
There might be a handful of pay-for options that support W98 for a time yet; I don't really know. In the absence of a resident AV working, once support for Avast 4.8 is withdrawn, your options start to become a little limited, I'm afraid, and basically involve not connecting the computer to the internet, or making sure that if you do connect you have a good backup strategy to restore the computer in the event of infection + NOT having any files (bank accounts/passwords/personal pictures etc) on the computer, and/or using a demand scanner like Clam periodically to remove infections.