OK, so I guess the first person to leave should report back here what happened.
To pay the fine at EZE, see this post, it explains it. I'd leave an extra hour at the airport to deal with it. You probably won't need that much time, but you never know.
http://baexpats.org/topic/31725-paying-overstay-fee-at-eze-at-night/
If you're not leaving from EZE, it's more complicated to pay the fine, this thread talks about that.
http://baexpats.org/topic/32309-can-you-still-pay-overstay-exit-fee-at-aeroparque-airport/
Supposedly you can now pay it online as well, not sure of the details.
I think you can return right away after paying the fine, but there is always the risk of triggering the dreaded 'perma-tourist' response at the border, where they could conceivably not let you back in.