Afkomen does have several rather different meanings. I'd say there are three principal ones, but perhaps there are more.
1) The most literal meaning, getting off of something. For instance, imagine a mother whose children are being wild and climbing on walls or other things they're not supposed to climb on, then the mother might say "Kom daar van af!" In the same vein, there was a huge hit in the Netherlands in the sixties called "Kom Van Dat Dak Af" (http://www.youtube.com/watch?v=ssfeP977NxI
- lots of youtube videos, really).
2) Colloquially, and perhaps counter-intuitively, to visit someone or something, or to drop by, in phrases like "Ik kom nog wel eens af", "Kom maar af wanneer je/ge zin hebt" and the like ("I'll drop by at some point", "Drop by whenever you feel like it").
3) Op iets/iemand afkomen, which I guess is really a phrasal verb of some sort, because the "op" is an essential part. This means, as you suspected, to come towards someone or something, literally or figuratively (for instance, "Daar zullen veel mensen op afkomen" = (literally) "Many people will come towards that", i.e. "Many people will be drawn to that", in the context of tourism, organising activities or the like).