You don't have to use H1, H2, H3 or H anything if you don't want to.
It doesn't really matter whether you are using CSS or not. What matters is the font size and the location of the header on the page. Google's on-page portion of their algorithm looks for larger font sizes on a page and the location of them and gives more weight to those words from an on-page standpoint. So if you put something in a large font at the top of your page, it works the same way as using an H1 on most sites using CSS. The H1 is just a shortcut for specifying and standardizing things like (font size, color, alignment, etc.) for headings across a website.
Remember, Google came into existence before everyone was using Cascading Style Sheets. When Google first came out, everyone I know of was coding their pages manually and specifying things like font sizes and font colors right in the HTML code on the page. Google could determine headings back then and they still can regardless of whether you use that H1 or not. (as long as you make it obvious for them)
To answer your question: Yes you should use headings on a page and that heading should include whatever keyword you are trying to rank for in most cases. That will make your page more relevant for that keyword. And relevancy is the most important ranking factor by far.
