I am looking for a word or phrase to describe someone who thinks their own problems should supersede everyone else's, and who is childish about it, too boot.
Specifically regarding females, specifically in East Asia, Princess sickness refers to exactly the personality traits you describe in the question.
Princess sickness, alternatively known as princess syndrome or princess disease [...] is a neologistic term used colloquially in some parts of Asia, notably Hong Kong, Macau, mainland China, South Korea, Japan and Taiwan, to describe a condition of narcissism, egocentrism and materialism in women, or "princess" behaviour. Conversely but less commonly, men with a similar outlook may be described as having "prince" sickness. - Wikipedia