We stayed in Panama City for a few days and I think that was enough for us. While the city seemed nicer and safer than other Central American capitals we have been to, it's not a budget friendly city. All the hotels/hostels seemed either cheap and really rough or okay but not worth the money for what you get. We stayed in the Las Vegas hotel and paid about $85 for a double room. And while it was adequate (air conditioning, wifi, fridge & microwave, good location), it seemed expensive compared to the rest of the country.
We went to Casco Viejo which was nice looking but seemed like a false front when you have already seen the real city. I do, however, recommend taking a cab (about $5 per person) out to Panama Viejo to see the ruins. It's free to wander the grounds and look at the beginnings of the city. We did not pay to get into the museum so it was a cheap outing. Also, we went out to the Panama Canal to see ships go through the locks. To save some money, I recommend only paying for the observation deck. Skip the museum and 3-D movie and watch the PBS special before you go.
While in the city we ate some great food but it was harder to find a good, cheap meal. Prepare to spend about $10-$20 per person on lunch or dinner in Panama City. We ate at Beirut and Caffe Pomodoro after having had our fill of Panamanian cuisine. Also, on average the national beers (Atlas, Balboa, and Panama beers) were $2.50 in the capital.