It doesn't help that the day-month order is different between English speaking countries. ;) Stick with the order you see. So:
"May 1st" is "May the first", or simply "May first".
"1 May" is "the first of May" (sometimes written as "1st May", but that's a little confusing - the first May of how many Mays?? Better to write "the 1st of May").