The Weather Underground website allows you to find the rainfall level from nearly anywhere in the world on any day as far back as 1945. The site provides a search bar for the city, ZIP code or airport code with a drop-down menu for month, day and year below.
The website provides a range of weather information for the specified time and place, including the rainfall level, measured in inches, in the Precipitation section. This section also includes details about the average monthly and yearly rainfall totals. Scrolling down to Hourly Weather History & Observations provides precipitation information, along with temperature, humidity, visibility and wind information, broken down by hour.