To me, 'bound to' and 'likely to' refer to probabilities, where 'bound to' is much more certain. 'Prone to' refers to a natural aptitude, (or in this case) inaptitude. A common usage is 'accident prone'. I would use it with 'feeling' rather than 'feel'. It suggests that the person is naturally nervous. Liable is similar, but has a meaning something like 'likely to suffer the consequences'. As John said, 'liable' and 'prone' definitely have negative connotations, which the first two don't. 'Apt' is not commonly used in that sense - sufficiently so that I can't be sure how it is normally used. It is much more commonly used to mean 'appropriate'. As John said, likely to and bound to both work in that sentence. Which one to use would depend on how likely you think it is that he would be nervous. 'Prone to' would work in the more general statement "He is prone to feeling nervous..."

There are slight differences. “Bound to” implies that he will almost certainly feel nervous, while the others don’t imply the same level of certainty. “Liable to” and “prone to” are best used for negative things, like nervousness, while the others are more neutral. Likely and prone both work well in this sentence because nervousness in an interview is a bad thing. But it you wanted to say “He is ________ do great in interviews”, you would not use “prone to” or “liable to.”

These terms mean the same thing (very possible or probable). They can be used interchangeably. (Obviously, you only need to use 'to' in the sentence once.)