There's a Hyatt right at the airport. You just go out towards parking, via the enclosed walkway, and you're there in a couple minutes. Other options are available and would have free shuttles.

I wouldn't recommend sleeping at the airport. There are no flights after about 11pm-midnight and before about 5am, at least that I'm aware of, so the airside terminal may even be completely cleared for those hours. Not sure how they work it. Maybe that's not practical to get everyone out given the inevitible delays and cancellations. But still, there's nowhere good to sleep. All the seating has armrests from what I remember. Any stretching out would have to be on the floor.

If you need to keep the cost down and aren't looking to "mattress run" for Hyatt along with your mileage run, you could Priceline a hotel for the PIT airport zone. I'll take a look at the lists, but I can't imagine anything you would get would not have a free shuttle.

EDIT: The highest available hotel quality is 3* in the PIT zone. Successful bids for April and May were running about $30-40 plus tax/fee (so probably a net of about $50, and bidding 2.5* probably won't cut that significantly). That's pretty good. Sometimes you even get the Hyatt that's right at the airport. The others should all be fine for a night's sleep as well. I don't think there's a truly hideous property in the 3* list, though I have to admit that I've never stayed in any of them given that I live here. Good luck.