How to quantify the numerical influence of categorical variables?
Yes you're talking about training a model, then doing sensitivity analysis to determine the numerical influence of each explanatory variable in isolation:

Some considerations/tips:


you said the three features are categorical. Presumably student_age is ordered categorical, and the other two are unordered. But you didn't say how many levels they each have (cardinality), or how many bins. You don't necessarily want to (say) bin student_age in bins of 1 year each. You might bin in say 2/3-year increments, or quantiles (see the R cut command), e.g. student_age: [0-17, 18-20, 21-23, 24-28, 29-34, 35+]. Similarly, if you have a long tail of smaller institutions with few exemplars, you might bin them together, or bin by country/ region/ size/ focus/ ranking.
what is the product of cardinality of each variable, vs the size of the dataset? Do you have 1 exemplar for each possible combination? >10? &lt;0.1? If you don't have that much data, you might prefer to use Linear Regression rather than NN, or at least a shallower NN rather than deep. With Linear Regression you can allow quadratic interaction terms, nonlinear terms etc - again 
(with Linear Regression you can directly see the influence just by directly reading the model coefficients e.g. Princeton: -1.4, Princeton.Age22_24: -3.6)
if you specifically want to use NN, see the lime package. Also see the many articles/tutorials on sensitivity analysis with NNs, and CrossValidated answers on sensitivity analysis. 
deep NNs suffer from a lack of explainability; you can view the many individual layer weights but it's near-impossible to picture how that translates into effect on output.