What legal or financial responsibility's do I have if I was just contacted 22 years later by a girl who claims to be my daughter

It would depend upon which state or country you are in, but in general, you have no legal obligation. Child support is payable to the custodial parent, not the child. If court-ordered, it is not normally retroactive (for a period preceding the institution of an action for child support), and ends when the child comes of age.

If you die without leaving a will, this person could, I guess, claim a share of your estate (if she is able to prove she is your child).