... Not precisely sure how to get the on/off specs that you require .......at this time.
However, whatever timer unit you select, you will probably have to utilize an interface of some sort between the timer and the pump. This would be either a relay, or else something called an FET.
The timer will provide the 'signal', and will activate the relay or FET, to allow current to flow through the pump motor.
..... Just a matter of looking around .....
Where did you get the diagram that you show in the post above?