Without any installation disks, I'd be tempted to run Carbon Copy Cloner on it to an external usb disk just in case for a backup.
I'm not exactly sure that you absolutely *need* to do a firmware update before loading Ubuntu (or any other Linux for that matter). At any rate, you can only do so from a hard disk that is currently running the older Mac-os.
I guess you'll have to wait for any sort of definite answer. I know quite a few early Macs out there got bricked by those trying to install OSX or using OSX utilities on them, when the installer would alter the video freqs in nvram, and then the older mac-os would not recognize the change without the firmware update, and shut off the monitor.
In Ubuntu's case, one has to go in and manually editi the xorg.conf file anyway, so this means that one HAS to use the "alternate" text-based installer, and possibly use a "rescue" mode, to edit the xorg.conf file before the screen shuts down until configured properly.
If you do decide to start upgrading the hardware, be sure to do a pmu/smu reset - anytime you change Apple hardware, this should be performed - not to be confused with resetting the pram - although that too is a good idea, but smu/pmu resets are nearly mandatory when adding/removing hardware. If you do go in there, you may want to treat yourself to a new motherboard battery - typically a Tadiran I believe.
You may want to look at your specific model to get more familiar with it:
http://www.everymac.com/systems/apple/imac/index-imac.html
To see what you might be getting into:
https://wiki.ubuntu.com/PowerPCKnownIssues#Blank%20screen%20on%20iMac%20G3
Also note that if you install a hard drive larger than oem size on those early imacs, you'll need to manually partition your disk during install to keep the root / partition under 8gb, typically 7.5gb or so since you don't want to be right on the edge of the partition, or if manufacturer's don't report the way they calculate the exact size.