Can I hand in an essay which I have written last year, but not as part of a course?

Could I just hand in this essay, or would I violate some rules with that?

If you wrote it as a personal exercise and it was never submitted to any course (i.e. never read), I can't see how anyone would notice the difference - provided you are right about the fact that it answers your assignment perfectly and you are not subconciously trying to make it fit into the assignment.

Should I just hand in this essay?

Aside from the rules, consider the teaching value of writing an essay (which you obviously already know, since you wrote one as a personal exercise!). If your workload allows it, you might find that re-working on that essay a year later is really instructive. Your perspective (on the content, but also on the writing level) probably changed in one year, and with the head-start of having a first version already, you could dig deeper into the subject and end up really learning a lot and producing an essay worth an amazing grade. Or, if you don't feel like reworking this one, maybe you could pick another subject altogether? The teaching goal of essays is usually to have you research a new subject; handing in your already written one, while being legal, would certainly not serve that purpose.

Could I just hand in this essay, or would I violate some rules with that?

Rules are specific to your institution, so we can't answer that for you. But I can't imagine why a rule would exist to prohibit this, and it would be very hard to implement.

Is this maybe already self-plagiarism?

No. Self-plagiarism is about claiming credit more than once for the same piece of work. If you've not submitted it anywhere before, you're fine.

Many institutions include in their definitions of plagiarism something like:

submitting a paper as original work in one course when the paper has already received credit in another course (unless prearranged with the professor).

(The above is from my own institution's Honor Code, as stated in the student handbook.) If you were at my institution, what you're proposing wouldn't violate the letter of the honor code since the work wasn't graded in another course. Your own institution probably has an honor code or a policy on "academic integrity"; it may or may not have similar language. As far as the "legality" of what you're proposing goes, your answer is probably in there somewhere.

That said, you probably still sense that your instructor might not look kindly on this—otherwise you wouldn't be here asking the question. If you were my student and you handed in work that you had written well before you even took my course, I would be somewhat disappointed. As @Kerkyra alludes to in their answer, the reason I assign essays & written work is to make you think about a subject in the context of my class, as part of the process of learning the course material. By presenting me with work whose preparation was completely separate from my course, you're depriving yourself of this opportunity to make deeper connections and understand the subject even better than you did before.

However, there could be other professors out there whose main goal in assigning a paper is simply to allow the student to demonstrate mastery of a particular subject; such a professor would probably be fine with you submitting your older work. Other professors might have different pedagogical reasons to reject the idea. So it probably boils down to the following: ask your professor whether this is acceptable. Depending on their pedagogical priorities in giving this assignment, several things might happen. They might say it's fine; they might say "I'd like you to substantially revise/rewrite it" (as suggested by @Kerkyra); they might say "actually, this paper wouldn't satisfying the requirements of this assignment, so you'll need to pick another topic"; they might say "I want everyone in the class to be writing about something they don't already know about"; or something else entirely. Ultimately, if you want to do something unusual for an assignment, it's better to ask permission of your professor than it is to risk a messy and protracted debate over its acceptability after the fact, when your grade is on the line.