Will you be in Hokkaido?
We toured there extensively this summer and cycled through several dozen tunnels: here our findings. They might apply to other parts of Japan.
Newest tunnels on big roads have wide side pavements where you can (and possibly are supposed to) cycle. Safety is not an issue but you might have to dismount to make it on the pavement and it might be only on one side of the tunnel. Lighting is not an issue, we found them very bright.
Slightly older or smaller tunnels still on big roads can have a smaller side pavement, sometime with a fence/chain separating it from the cars. It might be higher than the road or level. Safety is not an issue but if you have a wide load steering can be a bit tight. Visibility can be bad. Bring strong headlights.
Old tunnels on big roads can have a very narrow pavement where you can still push your bike, or you will have to share with cars. You will have to make the decision before getting in, as you might have a chain preventing you to switch. We found just one of those in Hokkaido (Otaru) and while safe, it was unpleasant. Visibility was poor. Surface was mud and sand.
Tunnels on secondary roads you will have to share with cars.
Things to keep in mind: you could have locals riding the other way on a narrow pavement without lights, often peasant/fishermen who are hopping across for their errands. Or pedestrians. Be very careful as you might see them at the last moment.
We also met several roadworks in tunnels, and a few times on the pavement, with the chain preventing you to get back on the road, but the crews were all extra kind and we sorted it out.
In costal/heavy rain areas you will find sand/mud banks at the entrance of the tunnel or sometimes in some points inside. Not necessarily visible and affecting your steering.
Concerning heavy breathing, tunnels (with one notable 10% exception on a side road on a dam) were never steeper than 6% and this helped. We did not find fumes to be extremely bad, and sometimes just cycling in towns on a slope at a traffic light, for instance, you'd get worst exhaust.
For planning tunnel location, length, and possible alternative routes, refer to "Touring Mapple" (which you should not ride in Japan without!)