The Arduino's a controller remember, not a power supply
This is true, but he specified his motor only used 120mA and the 3.3V regulator can source 150mA max. So why is that a no exactly?
I know it is not good to get close to your current capacity like that, but I would expect it to still work. And considering likely the rest of his circuity is using 5V regulator. And USB can source 500mA so he has no problem there (more current capacity yet if using an external supply).
And there is minimal harm in trying it. The worst thing that happens is the current draw is high enough to reset the Arduino. in which case it will become immediately obvious that motor will not work from Arduino 3.3V pin.
Can capacitor use for strengthen the current ?
Not to strengthen it, but it could be used as like a helping hand, so when the motor first starts to spin its initial current surge will come from capacitor not all from the 3V3 regulator. You would want a pretty large capacity capacitor though and it would need to be rated for the appropriate voltage.