I agree that it's definitely possible, but questionably enjoyable. You're talking about roughly 8 hours a day on the road (assuming you drive at the speed limit by the most direct route), after which your butt may hurt.
The midwest can be beautiful, and I think many visitors to the US (as well as plenty of otherwise-well-traveled Americans) miss out on the charms that the middle of our country offers. But if you're rushing through it, you don't exactly have much time to enjoy it. And US interstates can get pretty monotonous, particularly once you get to the Great Plains.
With gas and NYC car rental prices rather high, you might want to consider buying a one-way plane ticket from NYC to somewhere closer to your destination that has cheap airfare and rental cars, and then driving the rest. You'll probably break even on cost, and by cutting the drive in half, you'll give yourself more time to enjoy the remaining drive.
Whatever you do, I'd recommend getting a rental car with low mileage, comfy seats and satellite radio (broadcast radio in the US can get even more monotonous than the landscape). And enjoy!