"Austria-Hungary" didn't exist until 1867. The real succession is something like this, though it's a really really complicated thing. I had an Austrian history course in German in Vienna when I studied abroad so I probably know as much about this as a non-academic can. Believe me, you're saving yourself the flower of your youth by not looking too deeply into this.
The Holy Roman Empire was thought of as the continuation of the Roman Empire, but this was mostly a legal fiction since the empire was resurrected by and for Charlemagne hundreds of years after the end of the Western Empire, and even then the nature of this "empire" changed radically over the never thousand years. It basically was medieval Germany, with neighboring lands attached and sometimes parts of Italy. After 1648 the title of Holy roman Emperor was little more than another title that the powerful Habsburg dynasty had, worth more in prestige than power (which they had in spades). When Napoleon made himself Emperor in Rome, Francis II took the initiative to dissolve the Holy Roman Empire for reasons of politics and intrigue that I refuse to relearn.
After that, he was just the Emperor of Austria, which was pretty much all of central Europe. He and his successors were also the presidents of the German Confederation, which was in many ways a replacement for the HRE, but now honest about being German rather than Roman. In 1867, after the empire's defeat in the Austro-Prussian War ended any pretenses of Austria to someday uniting all of "Germany" under its rule, and the Hungarians demanded what can best be called a personal union between Hungary and "Austria," but there's really never been anything else quite like it. That lasted till the end of the First World War. I've left a lot out (like 1848) but this is enough.
The longest you could stretch out the life of the Roman Empire is 1806, and even that is built on sand. After that, the Habsburgs didn't claim imperial power in the tradition of Rome, but rather as a new, equally phony tradition of Austria. Calling the rulers of Austria-Hungary a direct continuation of the Roman Emperors is like calling Odoacer one.
I'm sure that's more than anyone would ever care to know.