It is unlikely that Bob's conduct would constitute disorderly conduct or disturbing the peace in the fact pattern you described and the police and the Mayor who ordered to police to act have probably violated Bob's constitutional rights.
While governments may adopt reasonable "time, place and manner" restrictions on free speech, and the inquiry as to whether a time, place and manner restriction is reasonable is a fact specific inquiry, the facts presented in this question are in the heartland of free speech protections and so an arrest is likely to violate a clearly established constitutional right.
Bob believes the mayor to be corrupt. The mayor is an ex police
officer. So Bob decides to protest in front of city hall at noon with
a sign and speaking loudly about his dislike for the mayor while
walking back and forth on a public sidewalk.
Bob is in a public place that has a long tradition of being a symbolic public forum for expressing grievances. He is articulating statements that he holds in good faith about a matter of public concern relating to a public figure. Noon is not a time at which there is a need to maintain quiet.
There is no indication in the question of any special facts that would modify the usual considerations in this fact pattern (i.e. the building is not on fire requiring fire departments to keep people clear, there isn't a Presidential motorcade nearby, there is no indication that there is an ordinance on the books in advance that attempts to reasonably accommodate protesters in the interest of some important interest (e.g. not interfering with security lines)).
Realistically, absent negative facts which are not mentioned, this looks like a strong case for a constitutional violation and a weak one for criminal liability.