Creating a power supply for a small machine
I'll list the basic design approach I would use:


Using TI NE555 timer as an example (other manufacturers make them too), it can operate from 4.5 V to 16 V power supply. Whether your existing power supply is suitable depends on whether it is regulated or not (since the power supply voltage for NE555 should not exceed 16 V, and must not exceed 18 V).
Configure NE555 as monostable, triggered by manual push switch.
Decoupling capacitor required close to NE555.
12 V relay, with contacts rated at minimum of 2 A inductive load (assuming your pump uses maximum of 1 A and allowing for 50% derating). This is used to switch the existing 12 V power supply to the existing pipette pump, for the duration timed by the NE555.
Low-power relay coils can be driven directly by NE555 output e.g. when sinking &lt;50 mA, NE555 output voltage drop is &lt;1 V. Therefore a 12 V relay with a minimum "pick up" coil voltage of &lt;11 V should work.
Relay coil freewheeling diode is required.
Choose NE555 timing components to fit desired duration of power to pipette pump.
Most general purpose relays will need wires soldering to them, in order to connect them to a breadboard. Those wires must not be too thin, to avoid excess voltage drop across the wires, and should be no longer than required. If you can't solder, then ready-made relays-on-PCBs-for-breadboard-use are available, but cost more and limit your choices.


If you need an extra regulated 12 V power supply (it's not clear to me from your question whether that is needed or not), then that would need to be added.