If I solved a already solved recent problem with a totally different method, can I communicate my solution in a journal? How likely is the acceptance of my paper?
Just to add to the above answer, make sure that you draw an explicit comparison between your method and the existing method in the paper and what are the advantages of your methods. You can write this as a separate section in the paper "Key Contributions". I am not sure that the paper will be accepted unless the new method has some distinct and sufficient advantage (because you have to answer the question of "why should anyone be bothered to use the new method").
Also, consider what are the shortcomings of the new approach and any approaches to mitigate them because reviewers will be interested in that too. If you don't have time to cover these shortcomings, you can justify how to mitigate them and write them as future works.
My advice is based on the fact that I recently submitted a paper (that got accepted) that proposed a new method to do something which had already been done by an existing method. But the advantage of my method was that it was more generic (the new method also works in scenarios where the old method doesn't). I basically showed this by referencing three prior works that had been unsuccessful in using the existing method in a new scenario that I was interested in and how the new method overcame the limitations of the old method. Some of the questions from reviewers focused on the applicability of the new method on future (yet unseen) scenarios.
Note: Take my answer with a grain of salt because my experience is only in engineering papers (specifically computer engineering/science). My advice may/may not be applicable to your area of research.