I remember what it felt like to be at the beck and call of the alcoholic in my life (my then-husband). My heart beat faster every time the phone rang, and as soon as I saw it was him calling, my heart ran into overdrive. I was always running to the rescue, helping him solve x,y,z problem, supporting him, listening to him, lending him cash, etc etc. It made me feel important, wanted, needed...
It was exhausting.
Very soon, I began to feel resentful even if he did thank me endlessly when he was sober. I started to feel like his caretaker, not his partner.
Have you considered reading Codependent No More, by Melodie Beattie? It might really help you in this situation.
Just because you are the only family your sister has doesn't mean you have to JUMP every time she calls. Some alkies are frightfully good at making everything seem like an emergency, and they become panicky when you don't respond right away. Your sister is in rehab; she's in good hands. If there's an emergency, the people who are caring for her are trained to handle it. If it's something beyond the scope of their expertise, then they'll call the people who can take care of things.
As much as this is a difficult time for both you and your sister, perhaps it's also time to put some healthy boundaries in place so that she begins to understand that she is not the sole reason for your existence and that you have many other fish to fry. That doesn't demean in any way the breadth and scope of your love for her; it simply keeps things healthy.