There are portable ductless mini-split units that sit on the floor, with an intake hose and a hose for moisture condensate, both of which would be routed through a panel in a window, similar to how a window unit is placed in an open window. These use the least amount of power. Have to be level, though, or condensate is going to leak.
The problem with trying to run an emergency system off of solar would be that you’d want a battery bank sufficient to run it through the night. By the time you get all that set up, it doesn’t make sense to only use it when the power’s out, you’d want it grid tied to reduce your electric bill. Also, your exposure to sun would need to be good, with a clear southern exposure not blocked by structures or trees or other plants.
There are numerous RV’s around with solar and battery banks now that are sufficient to run the air conditioning through the night, so it is possible, but those systems are also backed up by a generator typically. It’s not easy to answer. Here’s a description of a 2,000 watt solar system with a 1200 amp hour lithium ion battery bank on a 41’ Newmar motorhome that powers the air conditioning:
http://precisionrv.com/2017/08/24/2000-watt-1200-amp-hour-newmar-motorhome-solar-power-system/
The cheapest thing you could do would be to have a generator panel wired specifically to run a portable ductless mini-split or window unit, plus a few other things, a few lights, couple of electric outlets so that a smaller, much cheaper generator could definitely run it. $10,000 for a whole house generator to run your whole house just for an occasional power outage is excessive and really a luxury. You’re only wanting to make a room habitable, not power the entire house. Ask a reliable, trustworthy electrician what the cost would be. It’s hard to give a pat answer to any of this because what would need to be done would vary depending upon your needs and circumstance, but it would certainly be far less than $10,000.
We have a couple of those portable AC units. Use it just to cool the room we basically use. They have been real handy when the house units breakdown.
Lots of great suggestions on this thread... thanks Regulator...