I am probably wrong, but here are some ideas:
The obi device uses very little power, compared to a desktop computer. A medium-sized UPS should be able to keep it on for many hours before the battery runs down. (The same should be true for a router.) It should also filter out the "blips" and power spikes that might occur when the outside power goes down or comes back up. Finally, when it eventually runs out of battery, it should shut the obi device down cleanly, and bring it back up cleanly when the power is restored. So the UPS would be the first thing to check or upgrade if the obi device regularly stops functioning as a result of the public power going off and back on. Maybe the obi needs to be on its own UPS?
It sounds like power interruptions are enough to mess up the obi device, but not enough to reset it completely. That shouldn't happen if you have the obi plugged into a UPS (backup power supply), unless the power supply is shutting down (because of low battery) at almost exactly the same time as the external power is coming back up, so the obi ends up with a power "blip" instead of a nice steady power supply interrupted by a longish period of no power before power is restored.
If the UPS the obi is plugged in to is working properly, perhaps there is a phone or some other device connected to it that is transmitting a "spike" from the power failure to the obi, e.g. a powered phone connected to it that is not on the UPS? In other words, the spike might be coming from some other device and going directly to the obi, bypassing the filtering of the UPS? In that case, either disconnect the other device, or put it on a spike filter or UPS, or use one of those spike filters for the wire connecting it to the obi, e.g. the phone connection. Remember that most spike filters fail "on", meaning that after they protect you from one large enough spike, they become inoperative and cannot filter the next spike; but rather than staying "off", they remain "on with no protection", so you have no way of knowing if they are still protecting you. Your UPS should be less susceptible to this problem, or be able to give you a warning light if the spike protector died protecting you and isn't working any more.
Maybe you can configure the UPS to power down the "load" for some period of time (e.g. more than 10 seconds) before bringing it back up? Really, the UPS is supposed to prevent the problem you are describing. If the UPS is supplying power, it should be supplying even-enough power to prevent problems, especially during transitions and spikes on the public power supply. If it is shutting down too soon because of low battery, that may indicate the battery needs replacing or there are too many things plugged in to the UPS.
If the device the obi is plugged in to is really a router, it probably is getting its public IP address (WAN) from the internet provider (as a DHCP client), and giving the obi its "private" IP address (LAN) via DHCP (the router is the private DHCP server to the LAN). (The two DHCP functions are completely separate.) You can almost certainly set the IP it gives to the obi to the same value each time (perhaps by figuring out the obi's "MAC address" and tell the router to always give it the same private IP address as a "reservation"). That way you'll be confident of being able to connect to the obi if you can figure out the router's public IP address. Then if the router can use dynamic dns, you should be able to connect, if the devices are listening. (My obi's MAC address changed when I upgraded the firmware, so you'd need to watch for that. Otherwise the MAC address is always the same, so the router can match it to a private IP address.) The thing is, it sounds like the obi is off in never-never land, so even if you had all the IP addresses, it wouldn't let you in to the console, from which you could reset it.
Finally, as a last resort, you might need a remote power controller that you can connect to via the Internet. This would be another device inside the router (on its "private" or LAN side, the same side as the obi device). You could then use the router's "public" IP address (via dynamic DNS or some other method), and port forwarding, to connect through the router to the power controller and cycle the power off and back on for the obi device. Those devices aren't cheap, though one would pay for itself in savings on overseas calls, and aggravation. It sounds like that may be the only way to get the obi reset in the event of a power blip.
If you can connect a computer to the "private" (LAN) side of the router and leave it on all the time, you should be able to connect to it remotely. That might give you some additional options: you could tell the computer to play an obnoxious sound file repeatedly to notify someone to come reset the obi device; you might be able to connect a remote on/off device to the USB port, and cycle the power to the obi that way. You could turn on a camera and look around to see what's happening. It wouldn't need to be a terribly new computer to do those things. It would, however, be one more thing that might need to get cycled when a power failure occurs, and for which you'd need to have a UPS.
I don't know if there would be a device that would automatically leave the power on the obi off for several seconds, then turn it back on, as a result of a power failure. As I said, that might be a function available on the UPS device (mine does that sort of thing after it has shut down for low battery -- it always waits several long seconds after its power is restored before turning on the devices plugged in to it). Again, it's not a terribly complicated thing, just "wait a while before restoring power" -- but I don't recall seeing that available at the store in a separate device. So the remote shutoff device is probably your best bet.
I suppose another potential source of the problem could be that the UPS is supplying too "coarse" a replacement power, and that is causing the obi's power supply to fail. A UPS on battery supplies roughly the same kind of power as the utility, but it's not identical. (Think of the difference between a nice rounded sine wave, versus a squared-off wave.) That would suggest either replacing the obi's power supply (wall wart) or getting a different UPS that can generate power that the obi's power supply can handle. If the obi is on but not functioning because of bad power, the "reset itself on network failure" option you are suggesting would likely not work either.
Good luck! And let us know if you get something working.
