How can it be set up?
Governments pass laws (probably on a national or regional basis) to mandate clocks be set a different way, and when the time to change rolls around, people update things. Not too difficult to set up legally, but probably hard to enforce — see, for example, people in western China, who sometimes run their own clocks differently from official China time.
Should it be more efficient for the computer to synchronize and the internet to all run in the same time zone (same clock, same time, same number)?
No. If anything, it gets very slightly more complicated, since computer time registries have to keep track of past time zones and when the change was made in which places in order to properly list the times of past events. This already happens every time a country changes its time zone, so it's not a huge deal, but it doesn't reduce the amount of work computers working with time info have to do. (As Angel explained, computer hardware itself is synchronized to UTC anyway, so it wouldn't change anything internally.)
Do we encounter any hidden obstacles?
People are going to be pretty frustrated for a few generations as language has to adapt to change the meanings of "days", and for a while it'll be really infuriating to have everyone get used to the weekday changing while, for example, the sun is up and people are at work — where knowing the weekday might be important.
Also, again note how difficult this will be to enforce locally. People generally dislike arbitrary change that occurs only for the sake of whatever organizations lobbied for it.
In general, I ask a Single Time Zone on a 3D spatial spherical planet with a single Sun away from us like the Earth. One can ask if we have multiple suns or bright stars around the planet. In that case, the Single Time Zone really makes more sense, don't you agree?
There are two main configurations for planets in a binary star system: either the two stars are pretty close together and the planet goes around the pair (so using conventional solar time would still be pretty reasonable), or the planet goes around one and the other is far away (in which case star #2 changes its relative position in the sky over the course of months or years, which isn't something that a single time zone would really help people get used to; solar time using the closer, brighter star would work just fine). Triple or higher-order star systems are mostly just more complicated instances of these two cases.
However, a global time zone might be a good choice on a tidally locked planet where one side is locked into facing the sun, and thus the sun never appears to rise or set from anywhere on the planet. Since the day/night cycle happens at the same time everywhere (which is to say, it doesn't happen at all), the idea of staggering your time zones would be meaningless there.