Can I use this Anker Astro 3e to power my arduino through the usb?
Yes, the external battery pack shown should work fine as a power source for the Arduino: I use a similar (smaller) mobile phone supplementary battery pack for portable experiments with Arduino boards. 

Using the second USB port for powering the motors might be slightly more challenging, or not, for the following reasons:


If the battery pack defaults to 100 mA on the USB port and expects higher current to be negotiated by the client as per USB standards, the motors might not be happy with just 100 mA
From personal experience such battery packs are designed for minimum possible cost, thus may not have protective circuitry to cope with back EMF / spikes that motors would generate. You won't know till you try (and let the battery pack fry).
Finally, at the other extreme, the battery pack might not limit current at all, and if the motors demand more current than the pack can handle for extended periods, this might lead to an early demise of the battery pack.