It is likely that random forest would outperform a neural network for this task, especially if you don’t have large amount of training data. I think your choice of model is already the best bet: you will need to invest much less effort and probably get equal if not better results. Moreover, EXGBoost can plot the trained trees and give you insights on its rationale for the classification, something that neural networks would hide.
That said, if you are curious, the neural net architecture that I would use in your case is rather simple, just a standard multi-class classifier. Start with an input layer with a dimension corresponding to the total number of attributes and progressively shrink the dimension in each layer until you have an output layer with the same dimension as the total number of classes. Use a suitable loss function like categorical cross entropy to train your model. Use a softmax activation for your output layer, so you can interpret the output as the probability of each class.
The issue with even such a simple neural networks is that there is a huge number of choices to make, such as the number of layers, the dimension of each layer, the activation functions, the learning rate and number of epochs, the specific optimizer, whether to use batch normalization, etc. Starting simple, with a very small number of layers (even just a single hidden layer) and no fancy trick is probably the best way to quickly evaluate if it even makes sense to follow such path.
