Could a modern man live in the past and not betray himself? How important would immunity, appearance, language, etc. be? What problems would a person from the distant past experience in our world?
Only with careful preparation, and perhaps not even then.
We know lots of things, and we don't even think about it. How to cross a road safely. Where to expect the light switch in a room, and why putting fingers into the outlet is a bad idea. How to operate a smartphone. How to file a tax return.
There are lots of things we have "lost" because we don't need them any more. Cooking on a wood-burning stove. Riding a horse, or a horse-drawn buggy. Writing an entire letter by hand.
Any of those, or countless others, could betray the time traveler.
I think the differances either way are too vast to maintain a covert presence alone, at least initially. The Traveler would need a great deal of help- preferably from a full time minder. They would need to undergo an extensive period of training on language, dress, bearing and day to day living. One will have to get used to soap, while the other will have to adjust to going without. I would suggest that the Traveller starts with (after training) short visits with minimal interaction. Just go to observe and familiarize.