A standard alkaline 9V battery has a capacity rating of around 500mAh (milliAmp-hours). That rating is probably based on a current consumption of around 10mA. However, if the current consumption is higher, the apparent capacity drops. At 100mA, most 9V batteries have a capacity rating of about 300mAh, and at 500mA current draw, the apparent capacity drops down to 170mAh.
This means, if you are drawing 100mA of current constantly, a 9V battery will last (0.3Ah / 0.1A) = 3h.
A typical Arduino pro mini will draw current at around 20mA.
500mAh / 20mA = 20h
This is not even counting the motor and driver.
A standard alkaline AA battery has a nominal voltage of 1.5V and a capacity of about 2000mAh, and 3 of them will provide a voltage between 4.5V (fully charged
1.5V x 3) and 2.7V (fully discharged
0.9V x 3)
With that voltage range, you could operate the Arduino pro mini in 3.3V mode, with no linear regulator, at a reduced clock frequency, such as 8Mhz, which will reduce power consumption even more.
Being able to put the chip in power-saving mode, and periodically check for necessary activity will help even more.
See http://www.gammon.com.au/power for a description of many practical methods.
That said, the Arduino is going to be a pretty small contributor to power draw compared to a DC motor, so your overall run time will heavily depend on how often the motor is turned on.