It's quite a weird problem, at least for me... Following your approach you'll end up with $5^5$ classes, which is an intractable problem with typical Neural Networks. You can visist this paper, which shows a solution with mathematical rigour:

*  https://arxiv.org/pdf/1806.02507.pdf

I'll also expose three possible naive solutions, awating for people who have faced a similar problem:

* Convert your 5 columns into binary columns (3 bits for 5 levels) and treat the problem as a multi-label classification with 15 classes https://en.wikipedia.org/wiki/Multi-label_classification. 

* You can fit 5 multi-class models for each column (divide and conquer navie approach, it seems to be the state-of-the art phylosohpy)

* Treat your problem as a regression problem, but some questions arise: is '11111' truly close to '11112' in your dataset? i.e. are your labels a ordered set?

> So my question is if I should go on with the approach or are there better ideas? It is possible to use ANN to predict several properties (i.e. classes) at the same time?

Yes, an ANN can output more than one class for a sample (but will not work in such a large scale): 

* https://www.researchgate.net/publication/3297607_Multilabel_Neural_Networks_with_Applications_to_Functional_Genomics_and_Text_Categorization

* https://arxiv.org/pdf/1312.5419.pdf