Quite a lot, but perhaps not as much as you might think at first. There is also an important point to be made about your choice of the "Romans" as opposed to, say, "people in the first or second century CE".

Greek mathematics, which the Romans in principle had access to, had reached a great deal of sophistication by the 2nd century CE. A gifted practitioner familiar with Archimedes' methods (and certainly Archimedes himself, a couple of centuries earlier) would likely be able to solve many of the problems on a Calculus BC exam (although perhaps not those involving transcendental functions like exp or log), but the methods would look very different from the solutions produced by a calculus student. Archimedes, using geometric methods, was able to compute volumes and areas, approximate irrational numbers like pi, compute square roots and generally solve questions that might even give your typical calculus BC student some trouble.

Ptolemy's Almagest contains computations that are arguably even more intricate, and the practical computational problems he had to overcome are non-trivial even by modern standards. Look at this treatment of the Almagest using modern notation to get an idea.

With all that being said, the importance of the conceptual clarity that calculus and modern notation bring to these computations can't be overstated. Calculus was properly invented only in the 17th century, and has been the subject of successive refinements into the 20th century. Archimedes needed complicated, ad hoc geometric tricks to find the volumes of shapes he wanted to compute. The modern student merely needs to mechanically set up a triple integral and compute. Greeks and later philosophers grappled for centuries with the problems associated with infinite processes like series. Modern mathematical analysis defines clear rules for the manipulation of these objects, so that problems which were once challenging for geniuses are now routine, and can be explained to high schoolers. Ptolemy's work, for example, would have been greatly simplified had he had access to modern trigonometry. To convince yourself of that, you need only try to prove the trigonometric identities you know using only geometry (rather algebraic manipulations of sines and cosines, which Ptolemy did not know about).

Finally, a comment about the Romans, since I have concentrated on what the Greeks knew. Here's a quote by Cicero (from Tusculanae Disputationes):

With the Greeks geometry was regarded with the utmost respect and consequently none were held in greater honor than mathematicians, but we Roman have delimited the size of this art to the practical purposes of measuring and calculating.

This quote reflects the (still) generally accepted notion that for all their cultural and civilization's achievements in other fields, the Romans have done very little in the way of what would currently be considered mathematics. S. Cuomo has written about this (whether it is true, and why it might be so) in her book Ancient Mathematics (Ch. 5 and 6). She mostly attributes this to a cultural divide: mathematics for its own sake was less appealing to Romans. There were mathematicians at the height of the empire who were Roman citizens, but they tended to be ethnically Greek, or at least write in Greek. Some of the Romans' engineering and architectural feats most certainly required detailed calculations (this is something Vitruvius talks about in his Architecture), and indeed the methods of Roman land-surveyors (agrimensores) have been studied by historians of mathematics. Unfortunately, I do not know much about them, but it is clear that they would also have benefited greatly from modern trigonometry and notation to achieve greater efficiency, even if it their methods were sufficient for their purposes.