How to choose solution - Neural Neworks or Scikit-Learn/Numpy/Pandas?
There is no magic solution. You will have to try. The rule of thumb I would say is to try firstly classifiers which are less costly to train and deploy (i.e. Scikit-learn) then give it a try with NNs.

Anyways, for your problem, you may be interested to try the Genism library for topic modelling to extract what kind of topics are mentioned in an email.

Then spaCy is a great tool for NLP production tasks, there should be some text classification walkthroughs using it.

These are only a couple of ideas I am sure there are tons more. I believe the key thing is to understand which key indicators, features, words are responsible for an email to be labelled of a particular class.