Issues
What a wonderful question! There is so much to say about it!
It would really be quite difficult to bring either knowledge or items back to Classical times.
You will have difficulty with culture and the knowledge frame-set. For instance, Greek mathematics was closer to a religion – if you went to Pythagoras and started discussing mathematical issues he might want you killed as a punishment for revealing secrets. Remember he had Hippasus drowned for discovering irrational numbers. That’s if he understood you. It’s most likely he wouldn’t, even if you spoke perfect Greek.
To understand why, look at Aristotle’s explanation of force transmission. We think in, and use, Newtonian terms. He did not. As well as learning Latin and Greek, you would have to fit yourself into a frame of thought where ‘transmission of force’ translates as ‘propagation of species’, which sounds like biology. Classical physics operated with a fundamentally different approach to ours – objects moved or existed because it was ‘in their nature’ to do so. You can see that, without very specialist study of classical science and local cultural mores, you would be in the position of knowing more than them, but being unable to put it over.
I don’t think you could make a living calculating either - there was not the same stress on data for engineering, so there was less raw material for computers (the people, not the machines). There were, for instance, no standards for force measurement.
It is instructive to look at the experiences of people who lived in those days (or these days!) who have new and different ideas. They tended to get locked away, or worse. To succeed in a culture you have to fit in. And that means not being too far removed, or annoying the local establishment too much.
Even military hardware might be difficult to sell. Look at the submarine. The British didn’t encourage it because it was ‘unsporting’, and that was only a hundred years ago. Total War existed in classical times but was different – there would be no glory in using a machine gun from a hole in the ground, so a local military leader might reject modern weaponry for cultural reasons.
So perhaps, if there is so much difficulty working with a different culture, you might want to reject it and run on your own as a bandit? Modern weaponry would certainly give you an edge, but what would your modern immune system do? Smallpox, plague and cholera were much more common, possibly with other unknown diseases. I don’t know if modern vaccines would be effective against 2000-year-old diseases. And I suspect the modern bugs you bring with you would prove much more effective in killing the local people than any weapon you could carry.
Assuming you survive the local bugs, Art might be an effective cross-cultural item to trade. The only difficulty we have here is that classical art was a lot better than ours, so the interchange would be a bit one-way.