The standard Catholic apologetics argument from the Bible for apostolic succession is the selection of Matthias to succeed Judas (Acts 1:16-26). That includes taking note that the word for “office” in 1:20 is episkopos: the word for “bishop.” Thus, we have some sort of equation of apostles and bishops, which is necessary, for we believe that bishops are indeed the successors of (but not identical to) the apostles.
This very day, in dialogue with a Protestant on Facebook, I stumbled upon a “new” argument for succession from Scripture that had never occurred to me before in my 26 years of doing Catholic apologetics (I love when that happens!). I put “new” in quotes because I’m sure someone else has thought of this (“nothing new under the sun”), but for me it’s new, and I did come up with it on my own, even if others have taught it in the past. Dialogue and its intellectual challenge has a way of bringing about such wonderful discoveries.