Is it possible to charge a capacitors, and in return charge a battery?
A battery stores energy. If you want to charge from another source, then that has to supply that amount of energy, plus a bit more for losses.

How much energy does your power bank store? You say it's 10Ah. Unfortunately, you don't say at what voltage. Let's assume the manufacturer is trying to 'big up' their device, so we'll use the lowest plausible voltage of 3.7v. That's an energy of 10*3600*3.7 = 133kJ.

You've found an ultracap that can store 1300J. You would need around 100 of those if you could (a) extract all their energy and (b) move it to the power bank at 100% efficiency. With capacitors, you get 75% of their energy out with a 2:1 voltage swing, which keeps the converter design simple. Let's assume 80% converter efficiency. You would therefore need a stored energy of 133k / (0.75*0.8) = 222kJ, or 170 capacitors.

Batteries have a maximum charge rate specification. You can avoid damage to the battery by keeping your charge rate within specification.