Accuweather's Minutecast is usually accurate for rain to within 5 minutes and have used it to time my bike commute. It uses your specific location and if rain is passing over that specific spot, and when, over the next 4 hours.

Plenty have short term info about the current rain situation. You can check the radar too. Frankly just knowing that while it's clear here it's hammering it down not far away and it's been heading towards us this morning gives me a solid idea even if I'm not a meterologist.