The first time this happened to me was on my very first trip to Europe. We landed in Rome. I hadn't slept more than an hour or two on the overnight flight, and I'm not sure if that was a chunk of time or fifteen minutes here...twenty minutes there....
Our ride from the airport to the hotel woke me up a bit. Our driver clearly missed his calling as he should have been driving in high speed races which required quick reflexes. We drove up what I was certain must be an alley, but there were cars somehow fitting along the other side of this narrow passage. I'm pretty sure we almost took out a nun. I prayed like I've never prayed before...not so much for my safe arrival at the hotel as that we wouldn't kill anyone en route.
Upon arrival early in the day, our room wasn't ready. Okay. We locked our suitcases and left them behind the desk. The hotel furnished us with a map of the local area. We went outside and started walking. I've probably never felt prouder of myself when I managed to buy a snack and a drink at a local shop, mostly in really bad Italian. (Me with the really bad Italian. The locals were very kind and didn't laugh.)
We saw a few things I hadn't expected to see on my first day, including the Tiber. I could almost feel the ghost of my Latin teacher repeating sentences over my shoulder.
Long story short - I understand the need to officially arrive before settling into the exploration stage of things. I mark off things in my mind in stages. But I made it through that transition in spite of that. For me, a big part of it was saying to myself - when in Rome - and that kept my sense of humor going.
I have since decided that if something - like having a room ready - is significant to my travel comfort, I'll rearrange my plans to make that happen. I'll take a different flight. I'll book a hotel room starting the night before and let them know I'm arriving at 8 am. It's a comfort issue to me, and it's also been a live and learn issue to me. The more I travel, the more I see what works for me and mine. And then knowing that, I can plan better for the future.