How should one pray while on a long (around 15 hour) flight journey? How can one decide if the time for Salaah has commenced or finished?

In a journey, you can combine the prayers. So you can combine Zuhr and ‘Asr and shorten them; that is pray two rakahs for each. The same applies for Maghrib and ‘Isha. As for the exact timings, you can find out about the local Salaah timings of the places that you will be flying over and pray according to those. You can also research and get a general idea of the time when the plane would be in a particular area during the course of your journey. You can then pray accordingly.