Regarding public or private university, the Japanese company isnft really going to know your university in Spain anyway. And some of the most renown universities in Japan are private. (Others public)
Regarding what to study, all three specializations you are thinking about are quite similar, at least from a long term possibility of employment fields. I know that when you start university all these different curriculums seem to lead to totally different and quite predefined fields of work, but that isnft true. I would venture that most people who studied mathematics do NOT work as a mathematician. Just to give you an example, when I needed to decide what to study, I was thinking of pharmaceutical science. I had the opportunity for a 2 weeks stage in a pharmacy, and found it incredibly boring. So I reconsidered and studied Biotechnology instead. A few years into my working life I changed jobs and ended up in a Pharma company and have since been working in Pharma, surrounded by pharmacists (and people who studied biology, chemistry, veterinarian science, nursing, and even completely unrelated courses at university, such as law). This instead of what precisely you study, it is much more important to choose a general direction (and you seem to be clear on that) and then choose your first job wisely.
Now for moving to Japan it is quite unlikely that as a new graduate from abroad youfll be able to land a job in Japan (except English language teaching, but as PP said as non native speaker that can be tricky too). It is much easier to find a job in Japan once you have a solid (many years) experience in your field. OR ALTERNATIVELY to study and graduate IN Japan. If moving to Japan is your sole purpose in life, I would probably go that route. This would mean to do about 2 years of language school in Japan until you are at N1 level, then enroll into university in Japan and then find a job as newly graduate as all your classmates will do. I am not saying that I suggest you doing this, but if getting a job in Japan after graduation is your goal this might be the easiest way.
Lastly regarding learning Japanese, in my experience Japanese is easy at the beginning and gets more complicated over time. E.g. if you are learning German or Russian even to build a pretty simple sentence you need to know a good amount of grammar (eg declination and conjugation, been tensec) in Japanese to make your first correct sentences at A1/A2 level you hardly need any grammar, only 2 irregular verbs, not a lot of verb tenses, no declination, no conjugationc yes, you need to memorize a lot of words and reading/writing is a challenge in Japanese from the get go, but speaking is relatively easy. But unfortunately it doesnft stay that way. Both the vocabulary gets more and more complicated (in European languages, if you speak one of them, that isnft true, the more complex the topic the more likely that European languages use the same word stem). An other complication of advancing in Japanese is a lack of learners. A lot of people give up after A2, so there are very few courses available beyond that point. For this reason I think really learning Japanese can best be achieved in an official language school in Japan. So if you have the possibility during your university years to go to Japan eg for 6 months for a language course that would be good. It will probably just bring you to N4, but it also gives you a good impression of how life in Japan really is.
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