"I'll manage it" is less wordy. I think that "I'll be sure to do it" emphasizes the fact that you'll do a specific task for sure, and not the fact that you're sure you can manage to succeed at doing something (like studying for an exam).
I doubt that I've ever heard any of the sentences posed on this board as "does this sound natural", including those deemed to sound natural. Although I can imagine someone saying davis441's sentence, your alternative is certainly concise.
Thinking more about your original sentence, I can see the merit of a shorter sentence. For my own part, I'd be unlikely to combine "manage" and "be sure". If you are certain, use lawlin's suggestion. If you feel it will be difficult but likely, perhaps "I'll manage (to do it)".