Oh man, been there! Right before my cousin's wedding trip I somehow tweaked my neck moving furniture - worst timing ever. What actually helped me was a combo of things: ice first 20 mins on/off for the first day, then switch to heat after 48hrs (this order matters, don't skip ice first!). Also try sleeping with a rolled towel under your neck instead of a pillow, sounds weird but it takes pressure off instantly.

Traction stretches helped too - gently pulling your head straight up while sitting. Found a solid breakdown of these at pinched nerve in neck treatment at home remedies and exercises  which explained why the nerve gets compressed and what actually relieves it vs what makes it worse. Saved me from doing dumb stuff like cracking it myself lol.

Ibuprofen + elevation while resting, and avoid looking down at your phone. You'll be okay for the flight, just maybe grab a neck pillow at the airport. Hang in there!