How can I upgrade the firmware of an RCX 1.0 to an RCX 2.0?
The hardware of RCX 2.0 is identical to 1.5, and the only difference between these two and version 1.0 is the charging port which had been removed. So they're compatible and you can indeed use a newer firmware on an older brick.

You'll need the Mindstorms 2.5 SDK, which contains the latest official firmware. It's probably not on the LEGO site any more, but Philo has saved it for posterity on his site.

You may also want to use a compatible, but faster, firwmare from Dick Swan, you'll find it archived here.

As for uploading it to the RCX brick, I think there's indeed an option for it in BrickCC but I'm not too sure where, so I can't help much there. It's probably in the documentation somewhere.