If willing spent alot then Fujiya Hotel is the Best at Hakone ...build during Meiji Era plus is middle of town....walk to Train Station and walk to Museum...so near only....Not bad also with the Western and japan Mix architecture feeling.  BTW I discover hakone is famous Hot Spring Area in Japan...better stay at ryokan more Syok!!!

''Hakone Free Pass''  can get at Tokyo Shinjuku Station ....  Before Taking  Odakyu Odawara Line from Shinjuku to Hakone.

Can you do Hakone in 1/2 day? If you arrive at Hakone (I'm assuming you mean Hakone-Yomoto station) around noon, sure, I do it all the time. People do daytrips to Hakone all the time from Tokyo, leaving at say 9am and getting back at say 8pm. You, on the other hand, are already in Hakone and from what it sounds like are staying in Hakone for the night.

I've been to Hakone more than a couple of times. Sometimes I just stay in Odawara and make the daytrip up to Hakone, just because it is more convenient to get back to Tokyo from Odawara. I've also stayed at Moto-Hakone (which is near the lake), in Gora, up in Togendai by the ropeway etc. Where you stay depends on what you want out of your trip. I like Togendai or Ubako or Gora because of the view (my room opens up to view of Mt Fuji) but in terms of access, not so good i.e. more trains to get there.

Conversely, you may want to stay in Hakone-Yomoto (or even Odawara) for easier return to Tokyo. If I'm asked, I would pick Hakone-Yumoto for the 1st time traveler. It is nicely in between Odawara and say, the apex of the Hakone mountains. From Hakone-Yumoto then you take the switchback train up the mountain to Gora and perhaps then the cable train up to Sounzan and then the ropeway to Lake Aishi. The switchback train takes perhaps 1.5 hours to Gora, the cable car is maybe like 30 minutes to the top (be prepared, people run to these cable trains pushing every one out of the way! so much for polite Japanese myth!) and the ropeway, maybe 30 minutes end to end.

You may want to stop at Owakudani on the repeway to see what a sulfuric volcano looks like. Skip the black eggs - they taste just like eggs (duh!)

Then onto Lake Aishi and the boat ride. Maybe 30 minutes. Then you maybe to the Tokugawa checkpoint* and then it is time to go back. You'll probably take the bus back to Hakone-Yumoto.

* the checkpoint is fascinating if you like history as I do. Checkpoints all over Japan were the primary way the Tokugawa shogunate maintained power over all the other daimyos (the other major reason was forcing all the Hans[clans] to maintain expensive Edo quarters) for so long.
Mt Fuji is not in Hakone. You can, however, get an amazing view of it, particularly on the Hakone Ropeway. Also the reflection Fujisan casts on Lake Aishi is very nice and famous view.