Diamagnetic vs Paramagnetic
You're right that Cu0 has one unpaired electron in the 4s orbital. If you were to observe a single Cu0 atom in the gas phase, you would observe some paramagnetic behavior.

However, in metallic copper, the valence orbitals are engaged in metal-to-metal bonding. The 3d and 4s orbitals interact in constructive overlap with neighboring copper atoms to produce bonding molecular orbitals. 

When two 4s orbitals (each containing 1 unpaired electron) combine to form a sigma bonding MO, the electrons in that bonding MO become paired, per the image below.

Note: this MO diagram depicts the bonding in H2, but I'm just using it to illustrate the principle of partially filled atomic orbitals coming together to form a filled bonding MO



The process of deriving the MO diagram for a metallic solid is obviously more complicated than this, but the answer to your question is that a single Cu0 atom is paramagnetic, but Cu2 would be diamagnetic. By extension, metallic copper is effectively Cu$_\infty$ and also diamagnetic.

As you can tell, the process for determining whether an atom or ion is para/diamagnetic is a very different question than whether the bulk substance (metal) is para/diamagnetic. You can't do the latter by electron counting alone; you need to know more about bonding in the substance. As another example of this, you mention that lead is diamagnetic. Lead is another example of a paramagnetic atom (6p2; the 6p orbital contains two unpaired electrons) - but lead metal is diamagnetic because those electrons become paired in the bonding MOs.