Japanese airports are ridiculously efficient. I've gone through both Narita and Haneda a numerous times and it's never taken me more than 15 minutes to get through, even with an international flight. I can't quite remember how long they recommend for domestic flights, but I think it's 1 hour (i.e. 30 minutes before boarding).
Narita is right out for time to get there (like your scenario, it will take more than an hour to take a train there and make it to the check in desk). However, Haneda is only 19 minutes from Tokyo station by monorail. If you live in the more populated areas of Tokyo, it's probably not any more or less convenient than taking the Shinkansen.
Coincidentally, Fukuoka airport is only 3 km away from Hakata station (though you have to take a shuttle bus Edit: It's actually 2 stations away on the metro -- I didn't know this!). So in this example, it's really 6 of one half a dozen of the other. You end up in practically the same spot. I think this is one of the reasons why people fly to Fukuoka. In the other direction, I would probably fly to Sapporo if I needed to get there quickly, but would take the train if I was going anywhere in the Tohoku region (NE region of the main island). The main reason is that the Shinkansen stops at Shin-Hakodate-Hokuto and then you have to take a different train to Sapporo. When they finally have a line going all the way to Sapporo, I think it will be competitive because it takes an hour to get from Chitose airport to Sapporo station.
Hmm, many US airports recommend showing up a minimum of 2 hours early, and it takes me 1 hour to get to the airport...