As a rule of thumb, it´s usually better to have all your ASIs or feats for all your characters, so I will not advise ranger3 unless you got really really good stats from the start and you don´t want some feat. You already have extra attack as a monk so you do not really need more than 4 levels of ranger. So, one level or 4 (or 8, etc) if you want all your ASIs.
My choice would not be pick only one level of ranger tho, because the only non-RP reason I want levels of ranger for a monk would be hunter´s mark sinergy with flurry of blows (And to use it in a more efficient manner you want to go melee because is a bonus action to cast it like kensei´s shot, so you only get to do one of them per turn and it´s also a concentration spell) and that requires at least two levels of ranger. Sadly lvl 4 rangers do not improve lvl 3 spellcasting, you are stuck with 3 cantrips, 3 1st level spells.
That said, about taking levels of ranger, can I give you my personal opinion? As an elf you already have longsword, longbow, etc proficiency without taking any level in a martial class (And a kensei gets weapon proficiency with their chosen weapons automatically anyway).
And if you really want a fighting style and martial weapons proficiency, just pick one level of fighter instead of two of ranger. You have more fighting styles to choose from, also requires 13 dex and you could get 19 levels of monk, with all the ASIs and feats, included the level 19th one; all the monastic features of your subclass and empty body (plus Second wind, albeit at a very very low level).
About Horde breaker:
Horde Breaker: Once on each of your turns when you make a weapon Attack, you can make another Attack with the same weapon against a different creature that is within 5 feet of the original target and within range of your weapon.
The extra attack rocks, but
only works if the enemies are stacked together, less than 5ft. I do not know if you are playing a campaign with lots of mobs and choke points, but usually your targets are not all together, side by side, even more if you are attacking with a ranged weapon so the use will be situational at best.
You need 3 levels of ranger for that, forcing you to go 4 levels of ranger or lose one ASI/Feat. and also losing the last Monk order upgrade, Empty body and Perfect self of the monk class.
I am assuming you don´t want the rangers` favored enemy or natural explorer, voted worst class features 3 years in a row, for some reason for your build, of course, but if you want the ranger class to add flavour to your monk, go ahead, could be fun to roleplay that character concept.
