How chemists find out the electronic properties of elements (atomic numbers)
One could do it in the following way.


The atomic number $Z$ can be determined by examining the x-ray spectrum of an atom.
The frequency $f$ of the so-called K-alpha line is related to the atomic number $Z$ by Moseley's law
$$
    \sqrt f = k_1 \cdot \left(Z - k_2\right) \, .
$$ 
The atom is electrically neutral, so the number of electrons is obviously calculated from the atomic number.
And the number of neutrons is also simply to choose: the only requirement at this stage would be the correct atomic weight for the element in question.