You shouldn't need a syringe to fill a cart, if carts are actually what you're talking about. With carts, I would just take out the filler, cut it in about half, then put in enough drops of
juice to fill the cart about halfway. Put the filler back in and drop more
juice on until the top looks slightly slushy.
If you're actually talking about cartos (cartomizers) and they're the soft-tips, then just use a pair of tweezers or a straightened paperclip to pop the mouthpiece off, hold the carto at a slight angle and drip
juice onto the filler. Be careful not to get juice in the middle tube because it'll leak out of the threaded end. Start dripping just a couple drops at a time so they can soak in, then add a couple more until you've added around 20-25 drops. Easy.