In psychology, you have two types of measurement: either a trained observer judges the behavior of the subjects without them having to actively partake in any kind of test or experiment; or the test subjects fill in a test (self-report) or take part in an experiment.
Obviously the mood in a city cannot be re-created and measured in an experiment, but you still have the two other options:
1) If you want to estimate public mood without the cooperation of members of the public, all you need is an experienced observer. His observations will become more reliable, if he creates a scale of symptoms to watch out for (similar to the diagnostic tools in clinical psychology), instead of letting his gut-feeling decide. It would take a comparison of many observations over a long period of time to "normalize" or "standardize" these judgments. You could take objective data into account, like amount, speed and movement patterns of traffic, number of accidents, medical emergencies and their type (heart attack), public activities like street music, sunbathers etc. How these relate to certain moods I don't know, but a long term observation might show relations.
Pines and Maslach developed a matrix to calculate the resources
required to support a public event and use the audience profile for
assessment of crowd mood. Their model is two-fold. First, they use
descriptors to clearly identify separate groups (for example,
families, young adults, children, elderly, and rival factions).
Second, they attach a rating scale (1–5) to these groups. This rating
scale is used to grade the amount of verbal noise, physical movement
and overall audience participation. ...
Using this model to assess 35 events, Zeitz et
al found that crowd mood was an important factor in predicting medical
workload at a mass gathering event, although it did not significantly
affect the work of other emergency services, such as police or fire
and rescue. ...
Practical strategies to monitor and measure crowd mood
and type, along with the resultant behavior of a crowd, have received
limited attention. (http://dx.doi.org/10.1017/S1049023X12000155)
2) Similar to the example from Vilnius, Lithuania, given in a comment by Damien, or television audience measurement, you can select a representative sample from the population in question and measure their moods or relevant behaviors through self-report or objective measures (biophysiological measures, tracking mobile phone location etc.). Similar techniques have been used in a multitude of psychological experiments, from using mobile phone data to analyse friendship networks, to people reporting their behavior to the experimenters in real time using apps.
I am not aware of any research done on public moods in this way apart from the example from Vilnius, but I have not been looking very hard and I think something might be found, if you do a more thorough search. Relevant keywords might be "assessment, monitoring crowds, mass gatherings, mob psychology".