Is feature selection necessary?
Feature selection might be consider a stage to avoid.
You have to spend computation time in order to remove features and actually lose data and the methods that you have to do feature selection are not optimal since the problem is NP-Complete.
Using it doesn't sound like an offer that you cannot refuse.

So, what are the benefits of using it?


Many features and low samples/features ratio will introduce noise into your dataset. In such a case your classification algorithm are likely to overfit, and give you a false feeling of good performance.
Reducing the number of features will reduce the running time in the later stages. That in turn will enable you using algorithms of higher complexity, search for more hyper parameters or do more evaluations.
A smaller set of feature is more comprehendible to humans. That will enable you to focus on the main sources of predictability and do more exact feature engineering. If you will have to explain your model to a client, you are better presenting a model with 5 features than a model with 200 features. 


Now for your specific case:
I recommend that you'll begin in computing the correlations among the features and the concept. Computing correlations among all features is also informative.
Note that there are many types of useful correlations (e.g., Pearson, Mutual information) and many attributes that might effect them (e.g., sparseness, concept imbalance). Examining them instead of blindly go with a feature selection algorithm might save you plenty of time in the future.

I don't think that you will have a lot of running time problems with your dataset. However, your samples/features ratio isn't too high so you might benefit from feature selection.

Choose a classifier of low complexity(e.g., linear regression, a small decision tree) and use it as a benchmark. Try it on the full data set and on some dataset with a subset of the features. Such a benchmark will guid you in the use of feature selection. You will need such guidance since there are many options (e.g., the number of features to select, the feature selection algorithm) an since the goal is usually the predication and not the feature selection so feedback is at least one step away.