How does a Muslim pray the obligatory prayer when it is difficult for him to face the Qiblah and perform ablution (wudu) or dry ablution (tayammum), such as a traveler on an airplane who cannot find enough water for ablution and has no means for dry ablution? Does he pray to the best of his ability, as "Allah does not burden a soul beyond its capacity," or does he make up the prayer after his journey ends?

The ruling on praying in an airplane has been detailed in previous fatwas. As for someone who does not find sufficient water for ablution (wudu'):

If they find a small amount of water that is enough to wash some limbs for ablution, they must use it, based on the Prophet's (peace be upon him) saying: "And when I command you to do something, do as much of it as you can." If they do not find water, but find clean earth (tayammum (definition) material like dust or stones), they must perform tayammum, based on Allah's (the Exalted) saying: "And if you find no water, then take for yourselves clean earth, rubbing your faces and hands with it." If they find neither water nor clean earth, they should pray without purification due to necessity. The evidence for this is that the Companions prayed without ablution before the revelation of the tayammum verse, and the Prophet (peace be upon him) did not rebuke them or order them to repeat their prayers.