4. on another day in gotham city, the temperature is rising, the skies are getting cloudy, and some precipitation begins to fall. how would the barometer be changing?

Pressure reading in barometer will decrease

Explanation:

Barometer is used to measure atmospheric pressure which is nothing but the force exerted due to the weight of an air parcel above any land.

In case where this  pressure is low, air rises easily into the atmosphere where it cools down and condenses. Condensation produces clouds made up of water, ice and dust particles. After a certain time period these condensed water droplets fall as rainfall.

However, if pressure would have been high then the air and water vapor have never reached to a higher altitude where they could condense and form clouds.  

not sure if this is what you'r looking for nnew= nold