Can you play something called a 'Nega-Mage'?
While there does seem to be some third-party material in the world about a Nega-Mage, there is no RAW specific type to this. So, it's something you'd have to get. Based on RAW though, no. You can't.
What would be the closest concept to anti-magic fields?
What you're talking about isn't so much a 'class' of mages as it is a mage that has a ton of background count. Since I've never read the books you have, all I can do is quote to you what the game offers.
There's a concept in Shadowrun called a Background Count. When an awakened character walks into an area that has an affinity for anything other then his type of magic, he is subjected to this background count. Usually, you'll find background count hanging around affronts to life or magic. For example, if there's a sight of a car bombing that claims the life of a school bus full of orphans, that right there's going to have a background count. How high depends on your storyteller, of course.
This background count acts as interference for their ability to manipulate or otherwise use mana. This is because the mana is so twisted in that realm that they can't get as full of a grasp on it. In fourth edition, this resulted in their magic stat being lowered by one point for each point of background count. In fifth edition, it's a bit more lenient in that it only subjects the character to a dice pool penalty equal to the background count.
Sometimes, it's not just background count. Sometimes, you just walk into an area that happens to be another magic user's affinity or realm. At this point, they actually get bonuses to cast spells while you're getting negatives, since the area is attuned to them. You'll see this a lot with toxic mages or toxic shamans. They like to hang out in the fallout of nuclear blasts or places where toxic runoff happens.
What you're talking about is a mage (or character) that has a mobile background count. This is rare, but not unheard of. The best example I can give for that is a cyber zombie. These things are a horror-story of magic and technology where the soul of someone is forced into a cybernetic shell that doesn't hold enough essence for anything to exist there anymore. Because of how horrible the transformation usually is, it results in the cyber zombie walking around with a background count that interferes with magic.
Thanks to the way Shadowrun is kind of loose with it's ruling in some of these things, you can try and find a way to work that type of background count into a person and allow them to be a magical null: something where the mana streams are not able to work very well. But, that would have to work both ways, meaning that while the person can't be hit as easily with spells, they can't cast spells very well and other people's beneficial spells just slide right off of them. Plus, you should have other penalties, such as other living beings having this...bad feeling...about the character.