More runway capacity usually does mean airlines get more landing slots, so in theory there's less reason to hike prices just to manage demand, but realistically the savings won't show up overnight - airports and airlines tend to fill new capacity pretty quick before passing anything to passengers. The timeline's been dragging on for years, so don't expect major changes to your wallet or boarding experience anytime soon.
practical tip though: even with a second runway, flight prices still depend way more on when you book and how far ahead you travel than on airport capacity. book your flights on a Tuesday or Wednesday morning when fewer people are searching, and you'll probably save more than waiting for some theoretical runway-related price drop.