I'd like to add to the answer of @Thebluefish which suggests that Todd has enough medical experience to learn how to make antibiotics. I agree with this. Before going back in time, he could train himself in the procedure of making antibiotics. While the procedure is complex, it doesn't seem to depend on anything that was impossible in the 1300's. The main thing that's needed is the knowledge, and Todd has access to that.
The next problem is logistics. Todd can only make a limited amount of antibiotics, and he couldn't possibly distribute it to all of Eurasia. Moreover, he's a nobody in the eyes of the rulers, so it will be difficult for him to convince them to help him.
Therefore, he has to gradually earn trust. He should plan his visit for well before the actual outbreak of the Black Plague. He then needs to get himself something that will work as a laboratory, and start manufacturing a small amount of antibiotics. Once he has them, he needs to seek out people who are sick from a bacterial infection (not the from the actual Black Plague (Yersinia pestis), we are earlier than that in time). His first patients may be suffering from a lung infection or something like that. Many such diseases were thought incurable and often deadly at the time, and his "miracle pills" should certainly give him some attention. If he continues like this, it won't be long before he's invited to the ruler's court to create antibiotics for the people there. Before you know it, Todd will be the head of an educational institution which teaches how to manufacture and administer antibiotics.
Todd can also use this newfound position of respect and authority to spread ideas about hygiene, cats, etc. as discussed in other answers.
Todd needs to execute this plan in the capital of a large empire. That way the knowledge of how to manufacture and distribute the medicine will have spread throughout the empire and to the rest of the region before the outbreak of the Black Plague. Having antibiotics widely available will severely limit, if not entirely prevent the spread of the Black Plague.
Of course, Todd will have to start this plan very early. It will take years for this knowledge to spread. So Todd will have to spend his whole year teaching people about antibiotics. He will then have to leave, and it will be left to his students to continue the spread of the knowledge. This spread will have to go on for years after Todd has left. He will have to see the results of that work once he comes back to the present. (Those results, of course, will be huge, if Todd succeeds.)
There are, of course, many things that could go wrong, such as Todd being hanged for heresy even while trying to treat his very first patients. Or the ruler trying to keep the antibiotics secret to hide them from his enemies.
Also, I don't know if there is some part of the manufacturing process of antibiotics which absolutely requires modern technology (I skimmed the summary of the process, and the only thing I could see was the need to control temperatures, but that can be achieved using ice and tempered water.)
Other than that, it should be feasible for a story.