I don't think it will be too hard. On my Toyota's, I drain coolant from radiator and from an engine block drain. I make sure the heater it turned to highest heat setting. That's the only way you will empty what's in the heater core. I empty my coolant reservoir. Close everything back up and add Toyota Coolant
In your situation, I would drain all the coolant (I am assuming the civic has a block drain) and then add distilled water to fill up the system. I'd drive it with heater on full heat until it warms up good. Then I'd drain all of the distilled water and refill it again. Run it til it gets hot and then drain it all out. Then I would add the Honda blue coolant. In other words, I would do 2 rinses with distilled water.
If that seems more than you want to do, find a good mechanic who could flush the system for you.