I responded with the sentence "The prisoner will be afraid to die soon" (which is incorrect.) This sentence seemed to indicate to me that the thing which is happening soon is that the prisoner will die, and they are afraid of that happening, while the correct translation indicates that the the thing that will happen soon is that the prisoner will be afraid to die.
Which is the correct meaning of this sentence? How would the other be expressed in Klingon?
That is an accepted translation. It is possible that you had some other error?
English allows us to move the adverb around to give finer nuances of meaning, but in Klingon you can only put it at the very beginning. There is no way to differentiate (in a simple Klingon sentence) whether the tugh applies to the Hegh or the -vIp. This sentence can mean either and if context doesn't make it clear, the speaker might want to use more words and sentences to make it clearer. So for this exercise we have accepted three different placements of "soon", including the one you suggest.