routing number

Can you clarify what it is you're trying to do?

AFAIK, Apply Pay doesn't have a (public) routing number, but you should only need that if you're pushing money in/out of the account from somewhere else - e.g. you're trying to initiate a transfer (either debit or credit) FROM your checking account TO your Apple Pay account. In this case your bank will ask for the remote account number and routing number that you are trying to send the money to.

Apple Pay expects you to do it the other way around - namely initiate the transfer FROM within Apple Pay, nominating the remote account that you are trying to transfer funds to/from. In this way you don't need the account number or routing number associated with your Apple Pay account.

You cannot direct deposit to Apple Pay or Apple Cash. You need a bank. Apple Cash is not a bank and should not be used as one.