How can he know the times of prayer when he is on board the plane?
The one who is travelling by plane should pray when the time for prayer begins. In this case he can know it by the visible signs for the times of Maghrib, ‘Isha’ and Fajr. When the sun sets, then the time for Maghrib begins. When the red afterglow disappears, then the time for ‘Isha’ begins. When the dawn breaks, then the time for Fajr begins.
As for Zuhr and ‘Asr, you can work it out. If you know their timings in the countries over which you are flying, then you should delay the prayer for a short period and then pray, because time in the air is not like time on the ground. And you can join Zuhr and ‘Asr, and Maghrib and ‘Isha’, at the time of the earlier or later prayer, depending on what is easiest for you.
You could also ask the pilot, because he may have equipment that will enable him to work out the times of prayer in the air.
For more information on the times of prayer – on the ground – in most countries of the world, along with lines of latitude and longitude and the direction of the qiblah, you can visit the website of the Ministry for Awqaaf and Islamic affairs: http://prayer.al-islam.com/default.asp?l=eng and print out the prayer times for the country you are going to visit and over which you will travel on your journey as well.
It should be noted that if a person does not know the times but he tries to work it out as best he can, and prays, then his prayer is valid, unless he finds out that he prayed before the time for prayer began, in which case he should repeat it. But if he does not find out anything, then he does not have to do anything.
And Allaah knows best.