What percentage of people today would likely be descendants a man born 4,000 years ago?
To what degree (if any) of accuracy could his descendants be traced?


I don't really understand this question. To what degree can we trace its descendants given we know what from this man? Do we have a blood sample?


  What percentage of the global population is likely a descendant of the man?


If $N_0$ was the population size 4,000 years ago and $N$ is the population today, then the expected number of descendent of any given person is simply $\frac{N}{N_0}$. However, there is a huge variance and this is where things get more complicated.

As coalescent processes are geometrically distributed (well approximated by an exponential distribution if we assume that the population is large; more info in this answer), then the variance in coalescent times is equal to the expected coalescent time.

The expected coalescent time in a diploid population of constant size $N$ is $2N$. Of course, the assumption of constant population size doesn't hold for humans due to the recent demographic explosion. When the population size is increasing the genealogy looks like a star as coalescent events are rarer and therefore, it is more likely for a given individual living in the past to live descendant today (Excoffier et al. 2009). However, I don't know how to calculate coalescent time in expanding populations.