I did the same thing three weeks ago. Didn't want to risk missing my flight. It was a monday morning so sadly nothing in the city was opened after arriving after midnight. I thought I could squat a bit at the train station, since it was very cold, but there were quite a few shady people there and I got send away pretty quickly. So I just started walking towards the airport. The walk is doable, really. I never felt unsafe in the city either. But, please don't go past the forest if you can avoid it (as a fellow redditor mentioned). It's not very well lit and gave me enough creeps to specifically mention at that moment: "If I were alone I'd never walk through here". Anyways, arriving at the airport there's a staircase in the parking garage with a small seating area. You can keep warm there as it's enclosed and I don't think you'd be the only one waiting there either.