I am not aware of any venting requirements for a pantry - certainly not for any opening outside. If I were you I would close up the hole and seal it properly. For air exchange in the pantry, I'd put two vents in the pantry door - one low, one high, to allow air transfer.
Sounds like dehumidification might be the answer. The cause of the dank "basement" smell, is high humidity. Removing moisture from the air is a side effect of refrigeration, which is why the air conditioner helps. A dehumidifier is just like an air conditioner, except that the dehumidifier heats the air back up after cooling it and removing moisture.
You may be able to simply use a standalone dehumidifier, though then you'll have to dump the collection bucket. Another option, is to install a whole house (or whole apartment) dehumidifier. A whole house unit will be more expensive, but won't require dumping a bucket or water once a day.
If you have moisture problems in the basement, you'll want to sort those out. That should eliminate the cause of the foul air, and the need for removing it.
If you don't want to deal with the moisture problems, then you'll have to focus on removing the foul air and replacing it. This means drawing in outside air, conditioning it (heating, cooling, filter, humidify, dehumidify, etc.), and exhausting the foul air.
If you live in a mild climate where indoor and outdoor temperatures are similar, this shouldn't be that bad. However, if you live where indoor and outdoor temperatures are extremely different, you're going to be paying a lot to condition the air.
You'll want to look in to a heat recovery ventilation (HRV), and/or energy recovery ventilation system. As these systems will help reduce the amount of heating/cooling that will have to be added to the incoming air.
Drawing air from one dwelling unit into another, is not a proper solution. Air must be moved within a single dwelling unit, or between the inside and out.
Best Answer
I own a mansard and we are battling this issue. We had our roof replaced last year and the roofer that did it didn't even install 1/10th of the recommended ventilation. Last winter was brutal in the north and we sustained substantial water damage from condensation inside of our home. We are working with another roofer to get the proper edge venting installed. Think of a cold glass on a hot summer day, that is what the inside of my roof decking looked like last winter.
Do not skimp or you will be sorry and sustain a substantially decreased lifespan of your roof, not to mention you could incur damage to the interior of your home if you live in the north (hot inside of the roof + extreme cold outside = large amounts of condensation).
Make sure you have a proper vapor barrier as well. Loose fill insulation = more required ventilation. I would also consider a power vent with both thermostat and a humidistat. I am not a professional, but I am a Mechanical Engineer by education and since our issue, I have researched proper ventilation of mansards extensively. Go with a very reputable roofer that will stand behind their work if you have condensation issues.