"Today is the 43rd day of my English study plan." sounds best here

They could mean different things. #1 and #2 essentially say the same thing, though #2 is better as Judy says, but #3 might be different. Say you don't study at weekends, then Sat and Sun would be omitted from the count for the plan, as in #1 and #2, but are still counted in the calendar, which is what would be understood for #3, unless you state otherwise. Another colloquial way of saying #1 is "This is the 43rd day of my English studies". And of course it's "43rd" rather than "43th".