A real bed is a big purchase, and the night you get your keys is the worst time to make a big purchase. You are tired, you are stressed, and you will pick better once you have slept. So here is how to sleep well tonight without one.
Skip the air mattress
Air mattresses feel like a good idea and then deflate at 3am. A self-inflating sleeping pad rolls out in ninety seconds, does not leak, and packs away. It is the difference between waking up rested and waking up on the floor.
Two pillows, not one
One pillow is for sleeping. The second one is what makes it feel like a bed instead of a campsite. It is a small thing that does a lot for how the first night feels.
Assume the heat is off
Utilities take time to transfer. Bring a blanket and assume the apartment will be colder than you expect. Being cold is the fastest way to a bad first night.
Do all of that and you will wake up in a much better state to make the real decisions. When you are ready for an actual bed, the real-bed version of the First Night kit is a folding frame and a foam mattress that ships in a box.