Long story short, I have a Way of the Kensei Monk whom was raised in a magocracy, but he, himself, can't do magic. Because of that, he's utilized his intensive arcane studies to learn how to shut down spellcasters in combat.

Are there any notable feats besides Mage Slayer that are designed, whether specifically or circuitously, to basically punish spellcasters for breathing near me?

Hello, I suggest Spell Sniper. This feat allows you to learn a cantrip that requires an attack roll, such as Booming Blade or Green-Flame Blade. Additionally, your ranged spell attacks ignore half and three-quarters cover, giving you an advantage when targeting spellcasters from a distance.

You can get resistance to spells from a couple of subclasses and races - I think that should count if you are looking at a mage-slayer character

Resilient to gain proficiency in Saving Throws, though as a monk if you go high enough that won't matter. 

Shield Master, again, won't be useful as a monk unless the campaign ends quick but it would let you gain basically 1/2 of the Evasion feature you eventually get. 

You could reflavor Lucky to be more resistant to magic, and only use it when you're making saving throws against magical effects. 

An always good option would be to take Resilient - Wisdom. Monks become proficient in all saving throws at level 14, but not all games even reach that level. As a Monk you probably have high Wis, so getting proficiency in the saving throw means you have Proficiency to save against both DEX and WIS saving throws, which are the most common saving throws in the game.

Another one that could be useful but might not necessarily come up very often is Mobile. Spellcasters tend to avoid getting grouped up together, but if you've got multiple spellcasters concentrating on spells on opposite sides of the map, this can make it easier for you to reach them and break their concentration in a single turn without worrying about attacks of opportunity. This one is pretty niche, though, but on the plus side taking Mobile as a Monk is still fairly useful... all that mobility gets even more useful when you can just take a swing at anyone you're running past to avoid them getting a swing on you.

Although a potentially easier to use option for taking out spaced apart spellcasters would be to just take the Sharpshooter feat. Kensei Monks have solid ranged attacks, since you can take a Longbow as your Kensei weapon. You can even use Focused Aim to increase your odds of landing extra powerful +10 damage Sharpshooter attacks.

Unfortunately in a world ruled by magic, there are not many feats to counter it without using magic. My first thought was trying to find a way to cast or create an effect like counterspell or spell riposte, but there is no such thing in 5E that I know of. A solid feat might be something like Knight of the Sword from Dragonlance which drops a fear effect on a target of you melee attack. Maybe reaching into something like Eldritch Adept: Devil's Sight flavored as a natural ability in case they try darkness or the environment is gothic, haha. Going first always helps against mages, so any initiative increasing feat like Alert will help. Guile of the Cloud Giant helps to reduce damage and gives a free-ish misty step to help close distance so they can trigger your Mageslayer attack later.

Quote from markjack34 >>

Hello, I suggest Spell Sniper. This feat allows you to learn a cantrip that requires an attack roll, such as Booming Blade or Green-Flame Blade. Additionally, your ranged spell attacks ignore half and three-quarters cover, giving you an advantage when targeting spellcasters from a distance.

 

 Spell Sniper requires you to be able to cast at least one spell as a prerequisite.  And even if it didn't, it's odd to recommend a feat that grants a spell to a character who's whole shtick is that they can't cast spells.

A home brew magic shield would work pretty good.

Like a beholder shield.

I was a mage hunter investigator in a game and my GM built a magic sword as part of my organization.

Long Sword (Magic)

Counterspell 1/day (WIS is the casting stat)

Tracking Teleport 1/day (tracks a spellcaster that tries to flee with any sort of teleportation, by latching onto their slip stream and traveling with them. The bearer of the sword appears within 5' from the target they are following as if they were teleporting together.)

Could you get a ‘ring of verbal anti-magic’ that is just a spell storing ring with silence in it

Alert.

Can’t be surprised. Win initiative, stun/kill them before the casters turn.

the caster and their minions don’t have advantage on attacks against you if you can’t see because of weather or spell effects impairing your vision. Blinding or somehow having one way heavy obscurement isn’t an uncommon tactic for a caster.

Alert is good.

Get a spellwrought tattoo or 2, maybe ask your DM to homebrew it as permanent vs 1 use.  Silence, Blindness/Deafness come to mind.

A ring of spell storing for something higher, but would need to be refilled.

Ioun Stone of Reserve, and thematically that would be pretty badass floating around your monk's head.

That Tracking Teleport kinda ****s.