Classification/Regression Problem where Response Variable is Ordinal
You can treat this problem as multiclass classification and use classification algorithms to solve this problem. You can use any of the classification metrics like accuracy, precision, recall etc. 

Regression should be used only for the continuous data and continuous data is the one which can have infinite number of potential values within a given range. For example, for your problem the range is $1-10$, if it is continuous data, then it will have values like $1.04783,6.92838,8.2381,3.999,5.0$ etc. In this case you can opt for Regression algorithm.