How to reconcile with an old mentor after (unintentional) plagiarism accusations
Nobody is going to mention that the OP's old mentor put undergraduate work directly into a publication (i) without checking that it met professional-level standards of academic honesty and (ii) without informing the undergraduate that this was being done or that he was being added as a coauthor?  In my view, these transgressions are equally bad or a bit worse than including a few sentences without quotation marks from sources that have been cited in a survey paper.  

In my opinion there is a good chance that the mentor is embarrassed that these lapses of his own have been exposed, which complicates the OP's situation.  I am surprised that other answers seem determined to pin this all on the OP.  @Scott Seidman suggests that the OP contact the editor:


  I think an email to the editor involved, cc'ed to your mentor, taking responsibility (without qualifiers) and stating that none of the other authors were in a position to recognize or correct this prior submission,


In my view the mentor was in the position to recognize and correct academic cultural errors made by his undergraduate mentee, and since the OP was not involved in the preparation or submission of the manuscript in any way, the responsibility for this clearly lies at least partially (and I would argue, primarily) with the mentor.  Sending such an email to the editor could just call attention to his poor mentorship and supervision: i.e., it could actually make it worse.  (Pro tip: don't contact an editor about something unless you're sure all your coauthors are on board.  If there is any real doubt, clear it with them explicitly.  If someone is not returning your emails, you can't do this.)

The OP is not blameless in the situation: he made a mistake, and a mistake involving what sounds like a borderline instance of academic dishonesty.  He should apologize to his old mentor -- which he has done -- and move on.  I would recommend that the OP continue to treat his mentor like an old mentor: i.e., send him occasional emails keeping him up to date on his current academic life.  I would certainly not assume that someone never wants to hear from me again based on a single unanswered email.  But ultimately the mentor should know that he made mistakes as well, from a position of far more experience and responsibility.  It does not fall on the OP to take any kind of heroic or (especially) self-abnegating measures to remedy the situation.  One hopes that eventually these two people can move past one negative interaction amid many other good ones.