You don't need a syringe to fill cartomizers, but it helps with Boge cartos.
Regular size cartomizers take approximately 1 ml of eliquid, which equals ~20-25 drops, roughly. The first fill is critical--your carto needs to be fully saturated, or it will fry, but not overfilled to the point of leaking. You'll get a hang of it with some practice.
There are basically three methods that I use, depending on the cartomizer.
1) Syringe and needle; I use 18g sharp needle. Especially effective with Boge cartos, as those tend to be very tightly packed and are generally a PITA to fill properly the first time. The needle puts your juice exactly where it needs to go and the syringe lets you measure the amount of juice accurately. The drawbacks? It's work and not everyone owns a syringe. Gently insert the needle into the filler and slowly inject the juice; half on one side, half on the other side of the carto to ensure proper saturation. Afterwards you can refill by direct dripping from your bottle (method three).
Other options: an oiler bottle
or dripper caps:
Dripper Caps
2) Condom method, as shown in Phil's video. Works well with Boge cartos, but I find it messy--lots of wiping afterwards.
3) Dripping!!!! The best and easiest method for all cartos with loose filler (except for Boge). Just tilt and slowly turn your cartomizer as you drip down the walls into the filler, just like pouring beer.

Don't worry too much about the center tube; a little juice in there will prime the coil--just blow out the excess, especially if you're using automatic batteries!!!! A leaky carto will kill the switch in an auto battery.
When you're done, blow out the excess liquid from the center tube and wipe down the threads with a tissue.
Have fun.