Yes, you may use your old card until the new card arrives. Your new card will need to be activated by your iPhone and instructions for activating the card are included with the shipping material.
Your old card will cease to function once the new card is activated and you can request a mailer to recycle you old card.
Go to the link below, scroll down and select >> other devices recycle << in the drop down select >> Apple Card << and fill in address to send prepaid recycling materials.
https://www.apple.com/shop/trade-in