While it may be possible, I'm not sure if an update to include a whole online mode is likely. Nintendo hasn't really revealed the full potential of WiiConnect24 yet though, so it's impossible to know at this point in time.
I'm not sure how large the system updates are, but I expect anything along the lines of major game content to be much larger files. That would mean an extreamly long download time and then it would take up a fair share of you 512 MB of internal flash memory.
If you do the math it doesn't seem likely, unless the Wii is able to write extra data onto the game disks, which has never been mentioned by Nintendo.