All you can control is your own behavior. If you call someone, anyone, and they do not answer the phone, you leave a message and wait for them to get back to you or don't. If your sister does not want to keep in contact with you/your mother's condition, she does not have to; if something happens, she could miss out on being there and/or helping; that is her choice. There's nothing passive-aggressive about that that I can see. Your telling her how to manage her phone is out of line; she knows best how she wants to manage her phone.

It sounds like you are wishing others would do things the way you do them, have the thought that the way you do things is the only "correct" way to do them. You are not responsible for your mother, your sister, or anyone else; you are only responsible for yourself. If you are feeling resentful and as if your sister is not doing enough, I suggest you do less in the areas where you feel resentful and more that makes you feel good. You can only manage your own thoughts, feelings, behavior, life.