I have a severe allergy to anything new (carpet, paint, most flooring, etc.). I currently live on the 2nd floor of a 3 story apartment building. I just found out that the tenant above me will be moving out - so the apartment will be painted and prepared for the next tenant. My concern is that the VOC's from the new paint will pass through the flooring into my apartment. Is this something that occurs rapidly almost effortlessly, or something that is rather slow? I really appreciate your time.

No one here knows anything about the particular floor or paint involved in your situation. The best plan is to ask the building management to make sure the crew ventilates the apartment well after painting. You can arrange to be away that day so you don't smell it.

The floors seem to be a solid slab of concrete. Can gases pass through concrete? Are there any other possible routes the gasses could enter? I know some people say they can smell cigarette smoke from neighbors in apartments. It's not as simple as airing it out initially, I can't be in the same home as paint that is around 6 months old. I break out in severe hives.

Are there any other possible routes the gasses could enter?

Yes, there may be other possible routes, but it is not going through the concrete. Best plan is to ask them to air it out well and leave for a day or two. Have hive medicine handy upon return.

Install a small fan/vent to slightly pressurize your apartment. And seal any shared hvac ducts.

Wouldn't depressurizing the apartment promote infiltration from any unsealed floor penetration or crack?

It would, which it why I wrote "pressurize".

Sorry for some reason I read that as "depressurize". That's what I get for trying to browse the site using a smart phone.