As an undergrad student, how should I ask a professor for an opportunity to do research with them? I was wondering what I should do in this situation as an undergraduate student:

About a month ago, I applied for and did not get into a program that allows first-year students to work on research projects that professors put forward, with the intention of introducing students to research, which included a research stipend, housing, etc.

I was very interested in the professors topic of research, and was wondering if it would be okay for me to ask the professor for the opportunity to work with him essentially as an unpaid intern, as I would really like the opportunity to learn more about the topic.

Would he view this attempt to do research under him as desperate? Or would he appreciate the opportunity to have an extra research assistant?

I know that the answer would depend on the professor himself, and on my relationship with him, but how do professors in general feel about students coming up to them and asking to be a part of their research?
Absolutely, with no qualifications, approach the researcher and ask what it would take to do research in his/her lab. As a first-year undergraduate student, it's very likely that you don't have the knowledge to contribute to the projects. So approach humbly, asking what you can do to prepare yourself. Ask for suggestions for a review paper or a few research reports that would bring you up to speed. Ask if you can attend lab meetings as a fly-on-the-wall to absorb the controversies and decisions being made day-by-day. The very worst possible case is that you're told, "no." If so, you're already there. Next worst is that you invest your time and get only an education out of it. That's not so bad. You'll be positioned very very well for next year.

Good luck!