Take a look at the tagged documentation at
https://www.tumblr.com/docs/en/api/v2#tagged-method. You can provide a query string parameter "before" which will return posts posted before the provided date time. Then you'll likely have to filter the response of posts by their timestamp or date value and stop showing posts once it exceeds your limit.
For your example you can make a call with the before query param timestamp to May 30th, 2014 and iterate looking at each post date and stopping when it is less than May 1, 2014.