We rented a car at the airport and drove into Panama City.
Old City (Casco Viejo) is a 15 min drive from Panama City Center. You will have to drive through a poor neighborhood to get to the Old City.
We parked the car on the street for free (avenida A) and walk the Old City, which is about 8-10 blocks of history. The walk took about 2 hours. There are several expensive (for Panama) restaurants and shops.
Currently a heavy renovation is under way.