What ML/DL approach better suits this problem?
Yes, you can use DL network to solve this problem. It is a easy multi-classes classification task. And just a full connected network will do this work.

To construct the network I recommend using Keras which is easy to use. And before training the network, it is better to preprocess the data(standardizing the numeric column, and embedding the string column).

EDIT:

For the data normalisation you can reference this blog: Neural Network Data Normalization and Encoding, that show the basic methods and codes.