Matai matchil ha'tzom shel Tisha B'av?
Why don't you use את before the הצום?
You don't use את before הצום because הצום is not the object of the sentence. "The fast of Yom Kippur begins..."
Two answers are EXACULY the same
Thank you for giving me a little laughter today.
מתחיל in the end should also be accepted
It sounds very unnatural