Different faucets are a bit different so there is an element of guessing here but this is what I think I see.
I "think" that you will find that nut at the arrow will turn counterclock to unscrew, nothing to drop down, but with some form of rubbery O-ring to make a seal when either hot or cold go to the center and should go up and out the spout. As the spout turns it tends to wear this O-ring seal.
How bold is up to you and how hard it is to turn off the water. Bold is okay to take this part off but if you bump one of the handles while the spout is off, you can get a shower, so turning the water off at the source is safer! More trouble to turn off the water or more trouble to scrub down the kitchen?
My move would be to look for a time when I didn't need to use this faucet, take it apart and when going to town, take the part to match it up for fit on the new one. I might have to get my water for coffee from the bath. It's okay to take this part loose---just make sure the faucet handle stays closed if the water is on!
I found I could shoot water all the way to the ceiling if I was sleeping while working!
EDIT:
Like this maybe but they come in all kinds of different types, so taking the parts and spout to the store may be best and replace any parts of the seal at the time to avoid having to do it twice.
https://www.google.com/search?sa=X&t...63174209951683