When NASA says 5:50PM today, what location and/or timezone are they referring to?
This particular website (eyes.nasa.gov) is using your system's local time to tell you the closest approach in your timezone. So when it says 5:50 PM today, it is when your clock reads 5:50 PM. Just be warned, if you have your system set to something else or possibly if you're using a VPN, NASA's website may be working of the wrong information and tell you a time which is not correct for your current location.
How do I know if this asteroid will be visible at my location with the naked eye?
NASA's asteroid tracking website does not appear to provide any information relating to visibility or brightness. I'd suggest other websites such as in-the-sky.org. On the linked page it shows asteroids which may be visible from your location (which you can change or set properly), including the magnitude of the asteroid, the constellation to look at, and the time range to view.
The page for Asteroid 7482 tells you specific information about visible in your area. You might also use free software like Stellarium for figuring out where objects may be and their observability.
Although I would caution that for this specific asteroid, you may have trouble viewing it. You will need to be fairly far south and have a good telescope in a dark area as the apparent brightness doesn't seem to get above 10 or 11 magnitude