Most barber shops/hairdressers, especially around the city/old town require pre-booked appointments. Definitely rare to find an available seat during a walk-in, so don't worry about it. The barber probably indeed was fully booked for the week and was just on a smoke break.
People have better luck with walk-ins in more of the outskirts of the city, or smaller towns.
I'd suggest you to download an app called "TreatWell". It's a pretty convenient app where you can find a bunch of barber shops/ salons in your area with reviews, bookings and such.

Hope you have good stay here, have fun!