Say a person has been convicted in a federal court of a drug offense, such as conspiracy, and they flee from the prison sentence. Is there a statute of limitations where, after a certain amount of time, the person would no longer have to serve that sentence?

No. The statute of limitations pertains to the period of time within which a person must be charged with an offense. The principal theory behind the statute of limitations is that after a certain period of time, it isn't fair to make someone stand trial because witnesses who could support a defense may have moved or died or their whereabouts may be unknown. Even witnesses who are still around will find their memories of the crime have faded. Also, there needs to be closure at some point. None of these reasons apply to someone who has already been tried and sentenced. Once a person is convicted and sentenced, if he flees, the time he owes the government never goes away. If the feds catch him 30 years from now, he will still have to serve the sentence.