After posting a preprint on arXiv, or after an accepted paper appears online, how to bring other's attention to it?
(Unless one is already a big name in one's field, or one proves a long-standing and well-known open problem, I doubt that just waiting for things to happen is going to suffice.)
For sure one can give talks or present posters on relevant conferences. But are there any other methods of bringing it to the attention of others who might be interested and for whom it may be beneficial?
The question is both about "classical" methods and any relevant Internet tools.
In general, I would say no. Poster sessions and talks are the main way of sharing your research. You can try contacting the authors of any widely-known blogs in your field, and you can try sharing your research with any collaborators from other projects (i.e., ones who don't already know about it), but anything beyond that and you're venturing in the realm of being annoying.
There's one major exception that I can think of, and that's in the case where you're publishing a research analysis technique. In that situation, you can publish a toolkit that uses your method, and that may help speed up adoption of the technique. For example, in neuroscience, the SPM fMRI data analysis toolbox is very widely used. The group behind the toolbox came out a few years ago with a new set of techniques—DCM—for mapping brain activity, and incorporated those techniques into the toolbox. Because of this, many researchers now use this analysis tool. (Of course, this also means that you'll have many researchers using the tool incorrectly, and you may have to publish instructions on when and when not to apply the technique.)
JeffE pointed out another exception; if you're building on the results of another research group, it is definitely acceptable to contact the researchers in that group to let them know if your work. They're the most likely people to be interested in your work, and (arguably) they are in a position to give you the most useful feedback.