Tens of thousands of people fly from North America to Europe every day, and arrive the day after they left. While the majority of those people are U.S. citizens flying to countries for which they don't need visas or European citizens returning home, surely more than 100,000 every year need visas for their destinations. Some of those visas undoubtedly have validity that starts on the arrival date. You won't be the first person to have been in this situation. Relax.