Your arrival day is not Day One. It's Day Zero. I do not count it as part of my usable time in Europe. It's my day to just get acclimated, recover from geospatial displacement trauma, and get used to being on the other side of the world. I figure that I've just jumped to the other side of the earth, and that's not something to take lightly or pretend has no effect on me. It does.
I agree with those above who advocate putting more thought and planning into your flight details. Those who only shop based exclusively on price should not be surprised if other things suffer (a lot). I've taken that idea several steps further: if I'm going all the way to Europe (or Asia), which are flights from 10-14 hours long for me (US west coast) I now try to only fly business class. And not just any business class: one with a good lie-flat seat. And not from an east coast connection: I want a flight from my home (Seattle) or other west coast gateway non-stop all the way to Europe (if I have to connect, I will, but not from the midwest or east coast - because that makes the trans-Atlantic leg too short for a good stretch of sleep). For me, it's all about getting a decent bit of sleep on the way over, and after many tries, I've concluded that a flat, comfortable place to sleep at least a few hours without getting jostled is what I need. I know that's not an option for everyone, but it makes all the difference in the world to me and is well worth it. The difference is I now arrive rested, happy, with a big smile on my face and a spring in my step, and ready to hit the ground running (as opposed to arriving miserable, grumpy, exhausted, and physically sick, which is how I usually have been after prying myself out of a coach seat upon arrival in Europe). Essentially, that comfy seat "buys" me an extra day in Europe. My days in Europe are worth it to me, YMMV.
That said, my arrival day is still dedicated to being outside as much as the weather will allow, getting plenty of fresh air and some sun, staying active and walking around smiling. Check in and drop the bags if possible (if not, just drop the bags). Never, ever allow myself to "just take a short nap". Stay awake until a reasonable dinner time (slightly early is OK), have a light meal, then sleep deeply. The next day I'm on European time and it's all good from there.