I remember this so well from my Hell’s Kitchen apartment back when I lived in New York City! Because of the steam heat radiator, it had two temperatures: hot or freezing.
A mechanical air exchanger would be a major project that your landlord would have to install for the whole building … my guess is that’s not going to happen!
Probably the cheapest, most manageable solution would be an air purifier. Here’s what Natural Home had to say about them a couple years ago: A Guide to Air Filters.
Hope that helps some!