Does one always have to present new results when publishing a paper?
Not really. But author or contributors to the publication should present new and valuable scientific things to the future readers of their paper. I can classify these new things in a list bellow:
- An old method applied to solve a new problem;
- A new method applied to solve an old and a solved problem;
- A comparison of different old or new methods to solve a problem and discuss their accuracy or efficiency in solving problems by comparing their results;
- A review on different methods to solve problems.
As you can see, there is no need to always have new results in the papers, but papers should have something new for their readers. Of course if your reader wants to read about those two existing methods, he can simply find and read the original papers instead of your publications.
An author may find a newer method to solve a solved old problem but in a more efficient way. Or he may find a very hard solution which may not be that much accurate, but it is a unique and new approach to solve the problem.
In your researches of applying those old methods, probably you did some programming and soft computing or applied some computational methods which may be unique and of value to be published in form of a research paper.
In your case, the only thing that comes to my mind is that you may consult the professor whom you worked with to discuss whether your results are publishable or not. You have some choices of course:
Write a comparing paper on the methods you applied and discuss their accuracies by comparing their results. You may also present the computer programming you did during your research. Also, you can discuss the amount of time each method consumes for reaching same results accessed by applying either of methods. You may also do some statistical calculations on the data you have as a result of those existing methods, which may also be something of value to the readers in your field.
Work some more in the field of the researches you did during your undergraduate years and find some more results and data or find a newer method and publish your whole research in form of a paper.
[M]y question is that whether it would be possible to publish such
kind of research project in form of a formal paper or not[?]
If your manuscript is well processed and contains valuable information to be published, it has some chances to be published in a journal or presented in a conference.
But please note that not every paper is valuable for every journal. Some journals are very high reputable and have high levels of impact factors. So, please evaluate your paper and submit it to a journal that publishes the same level of papers as your own paper. A very low quality paper may not always have the chance to be published in a well known and high reputable journal.
Of course anyone may submit his manuscript to a journal or conference, but submitting a paper does not really mean that their editors and reviewers will accept it for publication.
[W]ould it affect in a positive way my scientific "career", since I
would be able to say that I started publishing studies early than the
usual (I mean, my knowledge tells me that publish stuff during
undergrad means "early")?
Having publications based on your research activities in the undergraduate years is a valuable point in a person's scientific CV, but please note that for applying to some scientific careers, you should have higher levels of research. As an instance, for one applying to a post-doc position, having a good PhD level dissertation and probably one or two publications (depending on the applicant's field) is of course of more value than his early undergraduate level research.
If you are seeking a scientific career after obtaining your undergraduate degree; your research activities may be of value, but it hugely depends on the taste of your employer and their expectations.
As a general answer to this part of your question, having some researches during your undergraduate years is a good point in your CV and will not hurt your academic background. But this is not all for a potential applicant to a scientific career; as you may have some skills, good letters of recommendations, good grades, etc. These are important too.
Moreover, would it be a positive fact to have papers publicized, even
if they don't have new results or anything unpublished?
I think that it would be positive to have a publication out of your undergraduate researches, as it may show that you are a potential and enthusiastic researcher and you are so encouraged in doing research.
But answering to whether it is positive or not to publish such results, hugely depends on your manuscript. If you simply copy and paste existing results without any new activities, seems not a positive activity to me. I prefer to read the original papers not a copy of them in a newer paper. If you do something new, even a very small thing is valuable and of course sounds so positive.
(Just joking but I call a person who just published existing results without anything new to read in his paper a professional copy-paster, not a researcher!)