Do you really want to have your computer on 24/7?
There are 1440 minutes in a day, one way to do it is to have an array of 1440*2 bytes where each bit in the bytes = a sprinkler (therefore 16 sprinklers can be handled). Every minute the Arduino gets the next two bytes (or 1.5 bytes in your case as you only need 12 bits) from the array and writes them to an IO port.
You will need either 12 spare IO pins or you can use a couple of shift registers or latches. You could do the whole thing with just 3 IO pins easily.
Now all you need is a way or initialising the array using a VB program or something on a PC and downloading it into the Arduino. Once that is done the Arduino doesn't need the PC until the next time you modify the timing program.
If you want the remote web access that's out of my area but the above is still appropriate and means that the Arduino is not dependant on the PC being up and running.
The solenoid valves are probably 24v AC, but if you use relays to control them it doesn't much matter what they are.
So, your total hardware can be 2 SIPO high-current shift registers (such as the TPIC68595) and 12 relays.