I would sooner die than (do that). Means I would prefer to die before I did that. (i.e. , I would rather die than do that)

It means: I don't want to do what you're asking me to do.

I would rather die... e.g. I would rather die than not eat ice cream.

It does not mean that the speaker will die soon. He or she will die in a short time [not "in a few time"] - but it does not mean that the speaker will die or wants to die. See Iangreen's answer below - it is correct. For example: You asked me to tell our boss that he is wrong, so I said, "I would rather die than tell him he is wrong."

that's mean I will die ' soon '