Bad datasheet?
You should be aware that the datasheet linked is most likely not the correct one for the valve you will receive if you buy through that link. Various user comments and even the official description suggest this.
Wrong architecture?
This is not the right type of valve for your application:
Note: This is not a gravity-feed solenoid. This means that you will
need enough water pressure to fully open the valve. Check the
datasheet below for exact details. It operates at a minimum of around
3 PSI
Your application (coffee maker) is most-likely gravity-fed. That means you have a tank of water and you let it out into your machinery for use driven solely by its own weight. With this valve, you will not be able to fully drain the tank. As the level gets lower, the pressure on the valve will decrease to the point that the valve won't open.
If you are using this valve you will need a pump on the tank-side of the system.
Wrong forum?
The detail of mechanical actuation and pressure systems is probably better handled on one of the mechanical engineering or physical sciences forums.
No, this transistor cannot be expected to do this job for long, if it can do it at all.
From the datasheet, look at the "On characteristics" on page 2.
First, its free air power rating is 0.625W, which means Vce had better be 1.25V or less at 500 ma.
Then, gain (hFE) is shown at different Vce voltages and currents. But significantly, not shown at Vce=1V and Ic=500mA, suggesting that the transistor is not rated to work under those conditions.
Finally, the CE saturation voltage is shown as 1.6V at 500mA, which exceeds the power rating shown above. You will be able to get away with that for a few seconds on a very low duty cycle. The MPS2222A would be a better choice, its Vce(sat) is shown as 1V here.
But...the above condition is achieved with Ib=50mA. This almost certainly exceeds the current available from your Arduino output pin.
If you are content to briefly overrate the device's power ratings, you could overcome the base current limitation using a second transistor as an emitter follower, to drive the base current you need.
(schematic editor isn't loading this morning, sorry)
Best Answer
It's most likely a motorised valve.
Figure 1. Motorised valve attached to water hose line.
An industrial version of a motorised valve would have feedback potentiometer so that the actual position of the valve could be confirmed and, perhaps, proportional control could be achieved. Alternatively limit switches could be fitted to stop the motor once full travel had been reached.
In this case and in the case of the garden hose timers you linked I suspect that the motor would be run for a time and if the valve doesn't move then "too bad". With five or ten cycles per day the run time would only be half a minute or so and battery life should be good.
simulate this circuit – Schematic created using CircuitLab
Figure 2. Simplified representation of control circuit for open / close control including optional limit switches. With SW1 and SW4 closed as shown the motor would run CW until the CW limit switch opened.
Figure 2 shows how a simple DC motor would be switched by a 'H-bridge' to effect forward and reverse motion. SW1 to SW4 would, in a practical circuit, be replaced by transistors (with protection diodes).