Running the AC is actually your best bet for a long drive, even if you're worried about fuel economy. The windows-cracked thing doesn't really work once you're on the highway - the aerodynamic drag at 60+ mph kills your mileage way more than the AC does. Your Civic's cooling system is designed to handle sustained use, so running it all day won't harm anything. Just make sure your AC is serviced and topped up before you leave, since a struggling system works way harder and burns more gas anyway.
The real fuel-economy win is in how you drive. Keep your speed steady and reasonable - dropping from 75 to 65 mph makes a huge difference. Use the AC on a moderate setting rather than blasting it on max; let it cool the cabin gradually instead of forcing it to work overtime. Crack the windows for maybe the first minute or two while parked to let hot air escape, then close them up once the AC kicks in. On a 12+ hour day, take breaks every couple hours anyway - you'll be more alert and it gives the engine a chance to settle.
One practical thing: if you're stopped at a rest area or gas station, leave the windows down for a bit before you get back in. Parking in shade whenever possible also means your AC doesn't start from an oven. The combination of reasonable driving speed, moderate AC use, and smart parking will keep you comfortable without tanking your fuel economy.