Three-year-olds are still working things through, and a minute seems like a year to them. She's old enough to be aware that the household is changing with the impending arrival of her brother, though she may not be able to talk about her feelings, and she may be saying, "I want to make sure I'm important around here!" (And this method may also have become a habit with her.)
However, her way of persuading you to do what she wants is not an acceptable one. She knows better ways of responding, since she singles out certain people for her hissy fit.
Here's an idea: You might start by letting her know that, from now on, crying for what she wants will not work - that when she cries you will NOT give her what she asks for. (Ask Daddy and Grandma to do this as well. Try to be all on the same team.)
Of course, she will test this to see if you mean it, and you'll have to mean it.
Instead, when she fusses, say, in your sternest (but not mad) voice, "Sit down RIGHT THERE." If she happens to do that, then finish up what you're doing for a few seconds; then sit her down on your lap, and say (more or less), "Did I tell you that if you fussed, you wouldn't get what you wanted?" Make sure she nods or says yes. "Did you fuss?" Same response. "And I'm not getting it for you, am I? That's the way it will be. You know the good way of asking, and when you ask that way, *sometimes* I will say no, but sometimes I might say yes. Would you like to try that way now?"
If she does, let her try. Then you can say in your best affected voice, "Ohhhhh, what a delightful way to ask me to help you! I would be enchanted to do what you request!" Make it funny.
The next time she rages, have her sit again, talk with her again, and set the timer for thirty seconds before she may ask you again the right way.
When Little Brother arrives, if you are working with him when Big Sister wants you, quickly find a way she can help, even if you have to make up something. She can hand you a clean diaper or bring the teddy bear or whatever. Talk to her about what you're doing, and what she was like when she was a new baby. Talking to your daughter while you're tending to the baby brings her into your circle, as it were, and she does have some of your attention.