In the given sentence
Für das bin ich zuständig.
is this sentence correct? I am skeptical because the object comes before the subject.
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The word order is fine, yes. "Dafür" would be used instead of "für das", though, regardless of the position.
Yes, it is one of the possible word orderings in German. With it, you put the focus to the object. Compare:
Ich bin zuständig für das
means "I am responsible for that". While
Für das bin ich zuständing
means "This is for what I am responsible".