Or if you were to take a taxi into town around 5am you'd be there by 5:45 at the latest, would cost around 1200NT. You could then take the express MRT back per the above poster's suggestion.
As to what you'd do at that time, there's probably not much going on except for the above-mentioned breakfast. You could certainly just walk around the main station area to get a look at the city. Also be aware that the airport MRT station is around a 5-7 minute walk west of the actual main station/breakfast shop (closer to Beimen station) so factor that in.
If Taipei is too much effort you could also enter Taiwan and go to the basement of Terminal 2 where there's a food court, or the 5th floor where there's an observation deck open from 06:30, convenience store and other restaurants (not sure if they'd all be open at that hour). There's actually one observation deck at each end of terminal 2 departures, one facing each runway.