Stick to your guns. She is going to cry & throw a tantrum (put it in a sippy cup with a lid; that will hopefully make the mess less messy to clean). You could try watering down the juice a lot and offering that instead, or maybe getting some of that flavored water (although, again, that is going to get expensive, too).
Give her only milk at meal times, and only after she's had
Toddlers, by preference, often want to drink their calories instead of eat them (hey, they were used to doing that for the first 90% of their life, right, either from a bottle or a breast, LOL). We have to be strong and do the right thing, even when it means real unpleasantness & frustration for us.
She might refuse to drink for a day, but she will come around. When all you offer is water, she'll eventually drink it.
Good luck!