Any way to test the pump directly? Looks like the system requires a battery, so it may be possible to test the pump from the battery directly (check pump specs on the label). If so, at least you would know if the pump works. If it works from battery, it is likely a matter of figuring out how to make the controller turn it on.
Looks like the rightmost button will turn the pump on and off. May have to hold it down for a second.
Make sure to follow the instructions regarding connecting the battery first. The controller has to see that first, and it will likely help to regulate the voltage. Panel first could fry things.
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Looks like it is not an expensive unit, so don't expect too much. Good luck with the troubleshooting.