Make arduino board controler for watering system with power bank
Is it possible? Yes.
Is it easy? Maybe.

A relay shield or similar would allow you to break the power connection from the solar panel to the power bank input, wait 5 seconds and then connect the power bank output to the pump.
Once you are done with the pump you could disconnect power from it and reconnect the charging power.

That is all fairly straight forward, maybe a dozen lines of code and some very simple wiring.

The issue is where does the arduino get power from? Or more accurately how does the arduino maintain power during the transitions between different power sources. How much of an issue this is depends a lot on how quickly the power pack transitions from charging to outputting and back and how you want the system to behave overnight or other times when there pump is idle but there isn't much sun light.

Also have you done a power budget? How much power does your pump take and how much power does your solar panel output under normal usage? Based on those two how long per day can you run the pump for assuming an ~80% efficient battery pack.