It means 'to be in a difficult situation' (in trouble, in a predicament, in a tight spot, in a tight corner, in a jam, in a fix).
I did notice that a couple of translators have rendered it 'in a hurry', but I think it was because there was a sense of urgency to get out of trouble.
Estoy apurado. = I'm in a hurry.
Tengo apuro. = I'm in a hurry.
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