Mac – Power up after UPS initiated shutdown

There is APC UPS Daemon can do exactly what you want:

Apcupsd can be used for power mangement and controlling most of APC's UPS models on Unix and Windows machines. Apcupsd works with most of APC's Smart-UPS models as well as most simple signalling models such a Back-UPS, and BackUPS-Office. During a power failure, apcupsd will inform the users about the power failure and that a shutdown may occur. If power is not restored, a system shutdown will follow when the battery is exhausted, a timeout (seconds) expires, or runtime expires based on internal APC calculations determined by power consumption rates. Apcupsd is licensed under the GPL version 2.

Your problem is likely a hardware problem, and is probably a short. Unless you disassembled the computer before using the rice method, it probably didn't help and you just ended up with a bunch of rice in the trash can. Also (and this is general advice for everyone), the first thing you should do in a situation like this is IMMEDIATELY disconnect all power to the computer, including removing the battery. Failure to do this will keep power going to the machine and dramatically increase the likelihood of a short.

Hardware failures do not necessarily happen all at once, and may not show up for a day or more. It sounds like this is what has probably happened to you. At the risk of being the bearer of more bad news by adding insult to injury, this type of damage also is not covered by Apple Care. Sorry you have to deal with this; best of luck to you in getting your machine back in order!