How feasible is it that we could see the Central Black Hole Sgr A* occlude one of it's closely orbiting stars? I'm thinking about the recent famous photo of a black hole, some months ago. It got me wondering if there isn't a way to more easily make a direct observation of these. 

We detect exoplanets in part by phenomenae that occur when they transit their star.

We also have excellent time lapse photography of numerous stars orbiting the Milky Way's central Black Hole (Sgr A*), with timescales quite amenable to human timescales.

Could we observe Sgr A* directly by observing one of these stars passing behind, or almost behind it?

The diameter of the BH is estimated at 60m km, more than 40x the diameter of our sun (1.4m km). At that distance, parallax related effects and relative sizes wouldn't be an issue, as the star and BH are virtually at the same distance both very far from earth. A star also wouldn't have to be occluded, just pass close enough behind, for gravitational lensing to be detected on that fraction of its orbit (but not other places).  

Additionally, a star could also be occluded by an accretion disk (if any), and its light affected in a way that would presumably show something of the nature of what it was passing behind. Admittedly Sgr A* may have little or no disk, but we might detect a disk of low enough density/size/activity not to be active, if a residual disk did in fact exist.

Are we likely to be able to see in time lapse photography, stars vanishing and reappearing behind an apparently empty patch of space, in this way? Has it been done?
I'm sure someone will write a longer/better answer, so I'll keep this short and just mention that when a black hole passes between the observer an a star, we usually say that it becomes gravitationally lensed rather than simply occulted. This can potentially provide much richer data than an occultation but requires more complex analysis. 

Optical/Infrared

It also takes heroic efforts to try to image near Sgr A* optically. You need to use infrared wavelengths because of dust and an array of telescopes using adaptive optics, combined interferometrically. See How did they make a video of the center of the galaxy, and what is it exactly that's flashing there?

If there was a dedicated, second Very Large Telescope array pointed at SgrA* all night every night, there is certainly the possibility that a gravitational microlensing event could occur, and an otherwise invisible star could suddenly brighten substantially as it passed directly behind Sgr A*. 

Are there candidates?

As far as I know there are no currently known stars closely orbiting Sgr A* that are likely to pass directly behind it. However, I suppose that there could be one so dim that it has not yet been observed, and a lensing event would be the first time it gets noticed, so this is certainly a possibility.

But lensing can brighten any background star, it would not have to be in a close orbit around Sgr A* at all. 

Radio

Gravitational microlensing works essentially the same at all electromagnetic wavelengths, so it can be observed in radio as well as it can optically.

However, individual stars tend to radiate most of their energy optically and are not so bright at radio wavelengths. It would be hard to pick out the radio radiation of a background star above the strong emission of heated plasma in SgrA*'s accretion disk.

To this end I have just asked How far have individual stars been seen by radio telescopes?

I think (but I'm not sure) that you'd have to pick a time of extreme quiet (little/no infall) to even try.



above: homemade GIF from ESA video ESOcast 173: First Successful Test of Einstein’s General Relativity Near Supermassive Black Hole