Reaching out to a stranger for company on a trip?

not weird at all!! you’ll probably get to know people on the trip regardless so it’ll all work out fine :)

I don't think it would be weird, especially since the class is online- a lot of people are probably in the same situation as you. Also, I would imagine that if your professor told you to talk to this person specifically, she's probably going alone as well. I've heard of companies for adults who are traveling solo and want to be able to meet up with other travelers in the same country, so they feel safer and can explore together. This is definitely not a unique situation, and there's nothing weird about it! You also have the common experience of being in the class together, so you'll have things to talk about if needed. Rome is incredible, I hope you have a great time!