These phrases are a little context dependent, but the short answer is no, they have slightly different meanings. "In the city" is a little more broad - it refers to any object that exists, or action that happens within the area of the city. "Across the city" is a little more specific - it either means something that exists on the other side of the city (Example - Her house is all the way across the city) or that a continuous action happens while traveling from one side of the city to the other (example: He drove across the city/The group has been performing across the city).
Given that one is more broad and the other is more specific, there are some situations where the meaning could overlap. Example: They performed in the city all year long vs. They performed across the city all year long. Even though the meanings are a little different in general, in this situation they overlap.