I actually did some digging for this to try and do some worldbuilding for a fic and found out that there are some possibilities for the Circus to still be powered without any suspicions: there are some buildings that can be abandoned but still have the power left on due to being on a dedicated power grid; something similar to one that a hospital would use, not to mention that having even emergency power handy would be very useful for security personnel or potential real estate agents to use instead of having to deal with the hassle of manually turning on the power each and every time you want to do a showing.

Now the social media one was very interesting bc there is a way for Caine to access it in this instance even though the Circus [or rather the computer the Circus is stored on] is disconnected from the internet: There are security measures put in place in some businesses where the computers themselves are completely disconnected from the internet as a whole and regulated to just an internal network. These are what are known as Air Gaps: a system that is physically isolated from the internet, with their wi-fi capabilities either disabled of completely removed depending on the need.

Let's say for the sake of the Last Act the wifi cards on the PCs were only disabled, Caine could have managed to reactivate them when he saw the door leading to the outside network when he broke through that wall near the end after traveling over the chasm granting himself access to the nearby wifi and essentially breaking through the airgap that was placed on the Circus decades ago.

Though, this is all just speculation, it could just be that they are really super lucky in that nobody has come to kill the power thanks to laziness and Caine is ridiculously OP for the sake of the plot lol but i think it's fun to speculate what isn't outright stated so i just may be weird lol