The answer is no, you just can't do it. You can make a character who is focused on anti-magic... but they are going to be a Magician or a Mystic Adept, and the reason for this is that spirits can go to the astral plane and as a mundane you can only do that with crazy drugs (and really, REALLY don't want to because you are helpless there).
If you just want to be a mundane who's good at fighting magicians/spirits, you can - the main things you won't have are the ability to protect other people from magic, or stop spirits from just going astral on you. That's not to say you'll be bad.
The core things you need are:
Arcane Arrester. This is the only thing that will stop Spirits of Man (or magicians) from WTFpwning you with combat spells. You can be a Fomori or Gnome (either is fine), or SURGE for it (less good, but if you really want to be a different metatype it's alright).
High Willpower (by "high" I mean "5, 6 if you are gnome)
Dareadrenaline
General "good at fighting" skills like anyone else, eg, body, armour, agility, reaction, etc. More specifically, you need to be rolling with weapons that hit armor/2, so your options are:
Various melee weapons/enhancements (stun clubs, shock gloves, etc). If you can use Martial Arts, this + Two-Weapon Style + Off-Hand-Training will let you kung-fu fight spirits to death in melee, which is actually quite useful as several spirit types are melee-oriented quite heavily. You CANNOT rely on this, though, because other spirit types will fly around zapping you.
Stick-n-shock
Gauss Rifle
White Phosphorus grenades (these are binary - Force 8+ spirits are immune, Force 7- spirits take damage).
You should have access to Guts - either a gland, or an autoinjector, or whatever. If you don't have Guts, 3/4 of the spirits out there will laugh at you running away Feared.
Optional:
Astral Hazing. It does let you hug spirits to death, but it has obvious drawbacks (eg, makes you super duper distinctive to anyone with astral vision, may annoy friendly mages).
Terrible ideas:
Attack of Will. It's a trap. Never use it. Don't try to be good at it.