Does it sound like it could work? What would happen when the relay will give power to the valve?
(I think I should get a power supply with more ampere, but I really don't know much about electric currents).
That solenoids label makes you think its a 12v 2Watt device, but the 2W is a product code.
I believe their current is approx 500ma at 12vdc so you would need a PSU of around 1A .
A 5V relay means its coil is rated at 5V, the relay contacts can switch the 12v without any problem.
You can buy some ready made 5v relay boards which interface directly to the Uno, or you can make your own reley driver with a simple transistor,base resistor and diode.