There are a number of journals where you might place an article on Ulysses, and your suitability for one journal or another is based mostly on the kind of argument you make. Is it a theoretical one? Historical? Just a close reading of the book?
One major journal that might, in principle, accept the piece would be Modernism/modernity. It’s a really sexy journal, though, so to be honest you’re not likely to get published there, especially if your name isn’t known in the academy, if you can’t use university letterhead for your cover letter, and if you don’t make a very theoretically advanced argument. Instead, you might consider Joyce Studies Annual, published by U Texas. They’ll be automatically interested in anything about Joyce, which at least gets you in the door. And they’re not as big a deal, which is an advantage for someone without a reputation.
Here are a few general notes on the field:
Your submission should be around 25–35 pages long, set in Times New Roman, 12-pt font, double-spaced, 1-inch margins all the way around, printed one-sided on plain white paper. Your last name and the page number should appear in the top-right header, except on the cover sheet, which should bear the paper title and your name and contact info, all in 12-pt TNR, center-aligned. If your paper is closer to 10 pages, which most undergraduate papers are, you might look at The Explicator, which is the only academic journal I know that prints short pieces. (N.B. They are mostly interested in new close readings, so might not be appropriate, depending on the scope of your argument.)
You should include a VERY short cover letter. All of my successful article submissions have had two-sentence cover letters: “I hope you’ll consider this article for publication in Journal X. Thanks for your attention, and I look forward to hearing from you.” If possible, print this letter (but not the paper itself) on university letterhead. But it looks like you’re no longer affiliated with a university. If you have related credentials (e.g., you’re a professional journalist) then make them known by including your position under the signature and/or using letterhead from your employer. But if you work in a factory, keep your profession vague. It’s hard enough getting published as a graduate student, let alone an independent scholar.
Perhaps most importantly, especially when writing about such a major figure as Joyce, your paper should thoroughly address recent scholarship on the issue you are treating. Especially because it’s Joyce, it’s very likely that there is a large amount of scholarship that relates to whatever it is you argue. (This would be different if you were writing about, say, the sociology of YouTube.) It’s important that your paper treat your argument’s intersections with both the major, seminal Joyce scholars of whatever period, and the current state of the field. That is, you should treat both the big critical guns, who may have written as early as the 1950s, and the scholarship from, say the past 15 years. This latter can be particularly challenging, but it’s important to show that you’re up on the scholarship and that you know how your argument fits in (or doesn’t). This may be less important if you’re sending an 8-page piece to The Explicator, but even they expect around 10 secondary references per article. (My most recent publication is 26 pages long and cites about 35 secondary sources.) In any case—and, again, especially because Joyce is so heavily read—you cannot expect to publish an academic piece that simply reads the novel alone.
One last question: I assume that you got a good grade on the paper in college, but did the professor suggest that it might be publishable? If so, you might ask her where she thought you might send it. If it’s been a few years, include a copy of the paper in your email, but generally profs are happy to give this kind of advice.
Let me know if you’ve got more questions. I can be more specific about journals, etc. if I know what kind of argument you make. Hope this helps!