https://www.sinks-taps.com/shop/hydr...tchen-tap.html
there's a small amount of water leaking from the vertical column from the join of the rotating tap arm. i guess either it is loose or maybe there's an internal seal that has gone bad.
can someone clue me in on how to open this thing up and fix it?
There are two failure modes, either the valve cartridge starts to leak, in which case it leaks all the time, or the seals between the body and the cartridge fail, in which case it only leaks while you're running the water (and for a while after).
I presume this isn't a rental.
Before you bother taking the top off, figure out exactly what mfr/model the tap is and check pricing on a replacement cartridge.
Sometimes it's cheaper to replace the whole thing.
Back before the brave new world era it was usually fairly trivial to stop a leaking faucet, but nowadays by the time the seals start to leak the whole thing is pretty much toast. (Even the expensive ones are sometimes made out of a cat hair/snot/bronze alloy.)
Apart from using a sledgehammer I think you need to start off by unscrewing the horizontal on/off lever (if too stiff for simple hand unscrewing use pliers but wrap the lever with a t-towel so as not to damage it with the teeth), then that should free-up the top circular end part...give it a go!
It's a rental, call your landlord.
If you feel like spending a few hours on your back in where the garbage normally lives, with your hands up in the space between the back face of the sink and the wall, shouting at the dog or your offspring/partner to pass you tools, before you call the landlord, then by all means do so.
He will send a plumber. The plumber will look at it briefly then replace the faucet, taking about 20 min to do so.
btw... If you do plan to spend some quality time fixing your landlord's plumbing, turn both the supply valves at the wall off before you start and make sure that they actually shut the supply off . Also flip the breakers supplying power to any outlets near the sink.