If you have a pump, best to use a mains timer - not a water computer.
Search for "mains timer" on ebay.

If you use a rainwater barrel, better use a filter - or drippers with big
holes in - or they'll get all gunked up.

Yep I'm worried too that the barrel wont create enough pressure. I'm using
gravity to lead the water from the barrel down to the "drip system" .. but I
need some kinda device to limit the flow of water to a finetuned amount of
water per hour.

What do you mean by a filter?