Welcome (from another BF newb) and hurray for your Continental! I've mostly wrench on early 70s 10 speeds, and IMO, you'll need to change at least the rear derailleur, more likely a significant portion of your drivetrain. For example, a RD that can hump a 39 tooth rear may require a new shifter, etc.
But this a good thing. You get a better functioning bike that has improved gear ratios.
Have you looked at changing the crankset to a compact 50-34 in a gear calculator? I'm guessing it's cheaper-easier than a custom freewheel and RD, even counting the bottom bracket adapter.
Gear Calculator
https://www.sheldonbrown.com/gear-calc.html
Last edited by BTinNYC; 01-03-22 at 08:50 AM.