I guess you mixed a couple of things here. Let me try and explain:
If you need personal
bank account - bank won't ask you for address proof. With Emirates ID you can open personal bank account from your phone and they will deliver your
debit card in a day or two to any address (they delivered me once in a mall as I was in the mall once the courier called me and I was not at the hotel at that moment so he brought it to the nearby mall).
For business banking, some banks may ask you for address proof for you personally (and other shareholders) and in that case you'll need to provide them with Ejari and some bansk (my case) will ask you to update new Ejari each year!
If you want to get a tax certificate, you'll need to spend 6months in UAE and provide bank statements (so they can see your salaries, expenses etc) and some other documents.
Please read this page for more details:
Issuance of Tax Residency Certificate
No need to open PO box for personal account!
Just use address of where you are located. Could be hotel address like for eg.: Hilton Dubai Jumeriah, apt 2206, JBR, Dubai and that's it...
It's just the address for the courier where to deliver your card. He can't leave that in PO box since it has to be personal delivery and you'll need to have Emirates ID (or Passport) with you. Some banks/couriers will ask for biometrics (to place finger in a fingerprint reader along with Emirates ID) so they can hand over you the card.