What I gathered from your post is that you do not want to get a hotel room but rather wander around in the city over night during your layover at Tokyo. You did not mention which airport you are arriving, NRT or HND. Few thoughts I have.

I say Tokyo is not 24 hrs. city. A lot of places close overnight. As you mentioned public transportations do not run overnight.

You mentioned you do not want to go to bars/clubs because you are 21 yrs. old solo traveler, but you will not have a problem wandering around the city of Tokyo alone overnight without place to stay? Beside that, kind of bars and clubs you may have in your mind is bars and clubs in the U.S., then bars/clubs in Tokyo will be more expensive than you may think. If you are not planning to get a hotel room because you are being cost conscious then bars/clubs may not save you money.

Those large onsen facilities which open 24 hrs. are good choice. There are places you can lie down, although will not be private place, and take a nap for short period. I understand that you are young 21 years old, but traveling overseas and spending all night in Tokyo without a place you can rest may be really pushing it…

If you are willing to pay for those 24 hrs. onsen facilities, which rates usually do go up during overnight period, why not get a room at business hotel so you will have a place to crash if you get tired.

Again Tokyo is not 24 hrs. city. Public transportation stop running around 12 midnight to 1 a.m. May found only few places open (the places do not cost you a lot). Your mobility area will be limited to few blocks of city area unless you are willing to pay for taxi, which I am guessing you do not want to pay for taxi.

Especially if you are arriving and departing from NRT, then the time you will be at Tokyo will not be prime time at Tokyo. Shopping at Shibuya will be out because those shops will be closed during the time you will be at Tokyo. Or you were referring Don Quijote when talking about shopping?

If you are at Shibuya and want to head to Tsukiji Fish Market for early morning tuna auction, then taxi is pretty much only option to get there.

Most temples do not allow outside visitors during overnight period. For example Senso-ji Temple main building closes at 5 p.m. and opens at 6 a.m. All those tourist places around Sanso-ji will be closed during overnight. Again, you have to take taxi to get to those places during overnight period.

The cost of taxi to get to places you have listed will not be different than business hotel room at Tokyo.

P.S.
You mentioned eating yakitori, I do not know source of your information about yakitori, you could have seen small yakitori places some travel media have shown located at Yurakucho or Shinjuku. Many of those small yakitori places do close by 12 midnight. After last train of the day there really is not much of customers, and usually the store owner (often store owner is the only worker in a small joint like that) try to catch the last train of the day to get home, hence have to close an hour to 30 min. before the last train.

Also, typical small yakitori joint is like small pub where people drink more than eat yakitori. Middle age salary men talking loud, once realize that you are gaijin (non-Japanese) those drunken middle age salary men may start bugging you to get you involved in drunken conversation.

You mentioned as 21 years old traveling solo, little bit scared going to a club/bar by yourself in a foreign country. Then I am thinking how can you be comfortable going to a small yakitori joint alone?

Last, Japan is known as one of safest place. However, that does not mean it is crime free. Wandering alone in a park alone at 3 a.m. in Tokyo?