Won't hold my breath

What does "I won't hold my breath" mean? Example someone told me "I know he promised you a good job, but I won't hold my breath for that one"

It means that you don't expect it to happen, so you won't bother to hold your breath in anticipation, or "await with bated breath." It's used as mildly sarcastic humor. Bated, or abated, breath, means technically with shallow or lessened breathing, not actually "held," although when used in connection with anticipation the distinction seems not very important.
SS

See what does the phrase 'with bated breath' mean?.