If you want to fight it just for the principle of the matter, you might have a chance depending on how cool the judge is. If I were you, I'd bring in proof of all your past passes and whatever daily ones you had for July and say when you bought the pass, you thought it was good for a month starting that day. If you bring in like a year's worth of passes, plus a pass that was bought for August before you got this ticket ... you might have a chance. To top it off you can then show the "no fare" and "invalid fare" mistake.
Chances are slim, but if you show you are a loyal DART customer and illustrate it as an honest mistake, you could get pity from the judge.