Sorry, but your submission has been removed because we don't allow hypothetical questions. If possible, please feel free to rephrase the question so that it does not call for such speculation, and resubmit Otherwise, this sort of thing is better suited for /r/HistoryWhatIf.
Gosh, big question. Short answer, it is impossible to say, there was five hundred years between the fall of the Republic and the fall of Rome, a lot can happen in five hundred years.
'Saving' the Republic would have required quite the effort in of itself, late Republican Rome was quite a mess, Caesar was neither the first nor the only person to try to 'overthrow' the Republic, and future destructive civil wars were not an impossibility.
The real reason this is essentially an impossible question to answer is that historians still argue over what caused the downfall of the Roman Empire. It is likely that the migrations of 'barbarians' would still have occurred, thus putting pressure on the frontier. No empire in history has managed to maintain consistent military or economic strength forever, and there's no reason to assume that a republican rome would have been better or worse at dealing with ebbs and flows of strength than the empire was. etc. etc.