Classification of all semisimple rings of a certain order I'd appreciate it if you tell me where to begin in order to solve this question: 
Classify (up to ring isomorphism) all semisimple rings of order 720.
Could the Wedderburn-Artin Structural Theorem be applicable?

Yes, you should definitely apply Artin-Wedderburn.
The thing you gain from knowing the ring is finite is that the ring will be a product of matrix rings over fields, since finite division rings are fields. Hopefully you know that all finite fields are of prime power order.
Now then, an n by n matrix ring over a field with q elements clearly has $q^{n^2}$ matrices. Start deducing what the possibilities are :)
