Can I renew my passport after giving the passport number to the airline?

I just booked tickets to Tokyo; the process included providing the airline with my passport number.

My passport expires three months after returning to the US. My understanding is that this won't be a problem for Japan (right?), but a number of other countries nearby require a passport that's valid for six months after exit.

Say I would like to renew my passport, just in case I decide to take a side trip, or just to avoid potential confusion with the six-month rule. Would it be an issue if I show a new passport, with a different passport number than I reserved the ticket with, when I show up at the airport? Should I contact the airline and update the number beforehand? Would I need to bring my old, canceled passport, or is a matching name enough?

(I am American with a US passport and, in case it matters, global entry and the accompanying TSA Pre-check)