Your biggest problem by far here is going to be getting other people to believe you.
I'm assuming this mystical tutor is from the future? Given that, surely he must know that there will be problems getting society to believe what they're told. However, we've already done solutions for that on this site: see How do you prove you're from the future?
Once you've got everyone believing what you tell them, you also need to be able to keep that belief: if everything you spout is complete nonsense and the principles don't work when tested, you'll pretty quickly get executed. This is Ancient Rome. The solution to this is to take it all slowly, much as the tutor should have done for you. Teach the basics first and prove that they work. Then move on to other more complicated things that rely on the basics.
As for what the greatest benefit to society would be, it's a mixed blessing. All this knowledge could easily bring back penicillin, agricultural success, good housing, food security - and overpopulation. Instead of the global population exploding in the 1800's around the Industrial era, it'll explode in the new industrial age: what we call Ancient Rome.
What that does to us, today, is unknowable. Essentially, our current situation is brought forward so it happens in c.100 AD instead of 2015 AD, giving us another 1900 years before we get to today. Scientists are predicting, as always, several different outcomes of overpopulation:
- The Earth becomes wildly overpopulated, food is scarce, people die, and the population drops (then the scenario repeats)
- We start implementing strict 2-child or even 1-child laws. This stops or reverses growth - for every couple, you need 2 children to sustain the population.
There are plenty of people making plenty of predictions, and nobody knows who's right.
Summary: it brings 2015 to the Romans. Great for them, perhaps not so great for us.