reaction between copper metal and aqueous sodium bicarbonate
First, reduction potentials.

Theory page
https://en.wikipedia.org/wiki/Reduction_potential?wprov=sfla1

Table of standard electrode potentials
https://en.wikipedia.org/wiki/Standard_electrode_potential_%28data_page%29?wprov=sfla1

Standard electrode potentials of sodium is -2.70V while copper (I) is 5.20V, single electron reduction of copper (II) has 1.59V, full reduction potential is 3.37V

Higher the reduction potential, more the tendency for the species itself to reduce. Reduction potential of Copper is much higher than Sodium.

So conclusion is no. 

Lets say if copper can replace sodium. By how much? Under room conditions, RT/F*lne is approximately 0.06V, so the amount of copper ions that will replace sodium will be orders of magnitude 71.5 to 115 times dilute. In other words, non existant.