I just did that same layover a couple of weeks ago, as my international flight to Lima got in around 11:30, and my Cuzco flight left at 5:40 AM. This is pretty common. I spent the time at the Airport terminal. There were probably between 50-70 people in the same situation, as this is a standard layover for folks going to Cuzco. We ended up getting pretty friendly and developing a cameraderie amongst us. It was, though subdued, almost a bit of a party atmosphere. We all started the night as perfect strangers, ended up comparing travel notes, playing cards, getting to know each other. It actually ended up being a pretty enjoyable time.

The fact is, there is no good alternative, unless you don't mind paying $200 to kill 3-4 hours at an overpriced airport Ramada Inn. I suggest you plan on staying at the airport and make an adventure out of it.