How to quickly check the purity of primary cultures of neurons

Ask the cells who they are. None of FACS, IF, or PCR can really define all the cell types in culture. 

Examine cell culture of neurons under a microscope or use a flow cytometry to isolate and quantify cell populations based on cell surface markers. 

Use a specific neuron markers. 

Find the most specific marker(s) for your neuronal type of interest and use immunostaining against DAPI.

The question is interesting, but what do you mean as quick?

If you have an advanced microscope or a flow cytometer
the characterization of cell surface marker of interest may be the best way.

So, Flow cytometry analysis is teh best and very fast way of doing that.

FLOWCYTOMETRY

It could be identified by Immunohistochemistry
Please see this reference 
Sahu MP, Nikkilä O, Lågas S, Kolehmainen S, Castrén E. Culturing primary neurons from rat hippocampus and cortex. Neuronal Signal. 2019 Apr 26;3(2):NS20180207. doi: 10.1042/NS20180207. PMID: 32714598; PMCID: PMC7363313.

Do Immunoflourescence markers upon culture (glial and non glial cells). To do at different time points corresponding to different "neuronal maturation". Hope this can help.

At first, the right sampling is important. after culture, you can use a positive control cell line and compare your cells with it in morphology, growth and proliferation style, and finally neuronal cell markers expression quantification.

biomarkers for FACS