Everything you need in terms of money in Copenhagen is a credit card. Also ApplePay and other smartphone payment applications are widely used. Many people only have their phone when it comes to payment. Make sure that you have a credit card that does not charge fees for payment in DKK since they will add up over time.
In Copenhagen you do not need any cash. Some places even do not accept cash as payment.
However, since you do not have MobilePay (Danish PayPal equivalent connected to a Danish phone number) you may need cash when going to the barber/hairdresser.