Hi Agonize, congrats on the new ProVari, I am sure you will love it. 18 to 20 drops is a rough guestimate since drop size can vary according to bottle. Dripping around the sides waiting for it to soak in is right. If you get it in the center tube, just blow the excess gently into a paper towel and wipe off the threads. I can usually get between 20 - 25 larger drops in a regular Boge carto. As for the syringe, that's probably the easiest method to fill a carto. Just suck the right amount of juice into the syringe, push the needle into the filler close the the sides of the carto all the way down to the bottom (be sure to avoid hitting the coils or the center tube, if you puncture the tube by mistake you could have a mess on your hands). Push a little of the juice into the filler, pull the syringe up a little, push a little more juice out, keep repeating in various places in the carto until there is no more juice in the syringe. Wait a couple of minute to be sure you achieved even distribution and vape away.
Cheers!