If a person cannot find water, is it better for him to delay his prayer or to pray at the beginning of the time?

That depends. It is preferable to delay the prayer until the
end of the time in two cases: 

1 – If he knows that he will find water, then it is
preferable to delay the prayer, although it was not said that this is
obligatory, because there is no certainty that he will find water.  

2 – If he thinks it most likely that he will find water, then
he should delay the prayer, because by doing so he is maintaining one of the
necessary conditions of the prayer, which is purifying oneself with water,
whereas praying at the beginning of the time is maintaining one of its
virtues only. 

In this case, delaying the prayer and purifying oneself with
water is preferable. 

Secondly, it is preferable to do the prayer at the beginning
of the time in three cases: 

1 – If he knows that he will not be able to find any water.

2 – If he thinks it most likely that he will not be able to
find any water.

3 – If he is uncertain and is not sure in either case.