How long does it take for the black hole at the center of our galaxy to make 1 full rotation?
The spin rate of the black hole at the centre of our Galaxy has not yet been established.
However, X-ray observations of gas falling into the supermassive black holes in many other galaxies appears to have established that most rotate at close to the maximum possible (see below, from here).
If we assume the 4 million solar mass black hole in our Galaxy is not unusual, then it too will have an event horizon moving at more than 60% of the speed of light.
As the spin increases, the event horizon shrinks, becoming half the Schwarzschild radius at the maximal spin. So for a 4 million solar mass black hole, the event horizon will be perhaps a little bigger than $GM/c^2 \simeq 6$ million km. If rotating near the speed of light, then an answer to your question would be a bit more than 2 minutes.
Measuring the spin of a black hole is a tough problem. There is a useful discussion of the problem in the answers to Coupling between galaxy spin and central black hole spin. It has been done, for example see this article in Nature, but only approximately and only in favourable circumstances.
Sadly Sagittarius A$^*$ doesn't lend itself to this sort of observation, so at the moment we have no idea how fast it is spinning.