A 5am arrival sounds intimidating in theory, but in practice it's quite manageable in my opinion, at least in Tokyo.

I suggest taking the Monorail to Shimbashi on the JR Yamanote Line (Monorail to Hamamatsucho and transfer to the JR Line, the transfer is really easy there). You can get off at Shimbashi and transfer to the Tokyo Metro.

As it will be quite early in the day you can buy a Metro Pass for about 600 yen and you can probably make very good use of it. It all depends on where your hostel is and what you want to see though, as a day pass for the JR lines might make more sense.

As RichardInSF suggested, you can drop off your bags at a coin locker if you don't want to go to your hostel. Here's an example of what you can fit in one of those handy 500yen lockers:



If you want to freshen up I suggest taking advantage of the early bird rates at a Spa like L'Aqua close to the Tokyo Dome so that you can have a dip in the hot pools or maybe have a quick nap.

With a Metro or JR day pass the entire city is at your disposal, and without bags you're free to do as much sightseeing as you want.

As you have an early start to the day, you can also take advantage and go to Yokohama or Kamakura, such a side trip would be a bit longer and therefore make 4pm like not such a long wait.

Good luck and enjoy !