I am not sure if it's possible,to open an account with the precaria, but I am "certain" that the bank will ask for your CUIL/CUIT.
I know a foreigner with a passport andd a CDI can open an account, but they also must have a CUIT/CUIL (according to what I was told recently at Banco Patagonia).
Since yuo are waiting for your DNI there would be no point in going to the trouble of getting a CDI, and you would have to still have valid 90 day visa to get it.
I believe your CUIL/CUIT will appear on your DNI but I don't know how you could know what it is until you actually have your DNI in hand.
I believe one member here recently got a CUIL with her preciaris by going to ANSES so she could start working before the arrival of her DNI. If your temporary residency is not based on work you could go to AFIP and ask about the CUIT.
I welcome any corrections of this and hope others can add to this answer.