a cheapo generic solenoid valve might be a solid way to start. There’s some more “professional” options but if it’s not getting buried underground and with a light duty cycle, not sure it’s worth the cost.
A flow meter might also be nifty for metrics or feedback or functionality confirmation.
For something like this, I’d go Pi over Arduino, simply for the network connectivity. Would be nice if you want a phone app or something to control it, or simply if you just want to hard-code a schedule (but still need NTP or something like that to get time of day).
For waterproofing I’d just look for any project enclosure that has some form of IP rating or at least a rubber gasket. Something like this. if it’s just gotta keep out the rain, shouldn’t be too hard.
Cable glands are my favorite way to get wires into and out of a box while retaining most of the waterproofness.