What can I do with my paper that is out of a work which contains no improvement of the methodology? I've started a research, and after replicating methods of other researchers (running experiments on my data), I've realised that I cannot add anything new to the field. The work contains no improvement of the methodology, I just used the established methods to analyse my data and I ended up in having some results. Consequently, I cannot publish a paper out of it (which I am almost done writing).

Nevertheless, the obtained results are new (applied to a specific problem), so they could be of interest. Furthermore, I don't want that my time and the possibly valuable results end up in the trash bin.

What can I do with it? Is arXiv suited for this?
You say that your work on methods is uninteresting, but that you believe the results to be valuable.  That's great!  You should write a paper about the results!

Sometimes people feel like the amount of manuscript text they produce for each aspect of their work should be proportional to the time and energy put into that aspect, e.g., if you spent 3/4 of your time working on the methods, then your text should be approximately 3/4 about the methods.  That is incorrect.  The amount of space something takes up in the text should be proportional to how interesting it is to the reader.  Sometimes you have to put in a lot of work on something that is very simple to explain, and that's OK.  Focus on the bits that will be interesting to the reader, and instead tell your friends the stories of how long and hard you worked on the "uninteresting" bits to get there.