I presume the answer is that it depends on the mass and size of the star and black hole and how they approach either other, but I was wondering if somebody could provide some rough bounds (e.g. hours vs thousands of years) theorized or based on historical observed data.
By "eating" I mean the time it takes a star to go through the event horizon of the black hole and potentially reach the singularity.
Answer
One of the main ways black holes are noticed is by looking at a solar system where the star appears to move as though it were a binary star system (e.i. two stars) when only one is seen. In these situations, depending on the distances, the black hole "feeds" off the original star, and a stream of the stellar plasma is slowly pealed off the star into the black hole.
This matter can sometimes form a very vivid accretion disk, that can be observed using telescopes (see Herbig–Haro object). This process can take a very long time, on the order of millions of years. However, of course, a rogue black hole could enter a star system head on and collide right with the sun and "suck it up," which would happen rather quickly (to an outside observer).
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