As nice as it is to have an solar panels to power your inverters, the main solution to this sort of problem is axing consumption. The main source of power to an inverter, without going nuts and spending ridiculous amounts of money on the initial outlay for solar panels, essentially has to be the mains. Going off one or two inexpensive panels, you'll still be draining your batteries - Just more slowly.
Given a reliably powered ISP, the secret to dealing with this sort of event on a regular basis is maintaining power to your router and/or modem. Reliable power to your ISP link and USB-chargeable portable devices that can be powered separately by dirt-cheap batteries & solar units intended solely for the purpose of providing USB power are enough to make anything bearable. Don't power anything you don't need. Run phones and tablets and netbooks on batteries in succession, and you can make it through 8-10 hours without power, until you can recharge your inverter & portable devices off the mains again.
A DS or a PSP's nice to have, too, for when the ISP goes down or your batteries for your router die.
Different problem from a prolonged outage, but it's a fun problem. Good excuse for a nap if you screw up. Good luck with cooking & food storage, though.