Refer to figure.
Vacuum-driven gyroscopic instruments (attitude indicator, directional gyro, and turn coordinator on some aircraft) use a stream of air directed onto the buckets or vanes of the gyroscope rotor to spin it at high RPM.

The engine-driven vacuum pump creates suction. This pressure differential causes outside air to flow through a filtered inlet, across the gyroscope rotor buckets, and into the low-pressure instrument case, from where it is discharged overboard by the vacuum pump. The flowing air jet impinging on the rotor buckets spins the gyroscope.

The vacuum pump therefore reduces pressure inside the system to create the suction that drives the airflow through the instruments. 

A suction gauge in the cockpit indicates the level of vacuum (typically 4.5 to 5.5 inHg below atmospheric) to confirm that adequate gyroscope spin-up is being maintained. 

Decrease the pressure within the instrument case → CORRECT. The vacuum pump evacuates air from the instrument case, creating a low-pressure region that draws filtered outside air through the system and across the gyroscope rotor buckets. The reduced pressure inside is what drives the airflow that spins the gyroscopes.

Ensure the pressure inside the instrument case is the same as atmospheric pressure → INCORRECT. If the pressure inside equalled atmospheric, there would be no pressure differential and no airflow through the system. The gyroscope rotors would not spin. The entire operating principle depends on maintaining a pressure differential between the inside of the instrument case (lower) and the outside atmosphere (higher).

Increase the pressure within the instrument case → INCORRECT. Increasing the pressure inside the instrument case is the opposite of what the vacuum pump does. A pump that increased internal pressure would be a pressure (positive pressure) system rather than a vacuum (negative pressure) system.