In between several days each in Tokyo and Koyto, I'm planning a 1-2 night stay in Hakone, estimated dates November 13-14.

In the Hakone area I'm most interested in walking the Old Tokaido Road, being in nature, a ryokan/onsen experience, and having some chill time between the busyness of Tokyo and what I'm sure will be the crowds of Kyoto. Other attractions of Hakone are of interest but not necessarily must-dos. A few questions regarding ryokans:

• I will be traveling alone. As of now, I'm planning to stay in Hakone for 1-2 weeknights. Will I have a problem booking a room in a ryokan for just one person?
• I don't like any fish/seafood and I'm just generally picky about meat, so I'm thinking a non-seafood or vegetarian meal option would work best for me. Is that possible at ryokans? I know you generally can't customize meals in Japan. (I don't mind dishes cooked with fish sauce or other seafood products, as long as it doesn't taste fishy to me, which I realize is pretty subjective.)
• Ideally I'd love to stay at a ryokan with a gorgeous outdoor onsen that feels like you're in the middle of a garden. My budget isn't set in stone, but I'm thinking $700 for a two-night stay, although I can cut it down to just one night (keeping the $700 budget) if I need to. The onsen doesn't need to be in my room, but ideally it would have times available to book privately. Not a deal-breaker if it doesn't though; I'd prefer a beautiful single-sex shared onsen to a private tub in what looks like a normal bathroom.

Any specific ryokan recommendations that meet the criteria above?

I'm also open to recommendations for an onsen town other than Hakone. It would need to be somewhere I could get to fairly easily between Tokyo and Kyoto, that would accept a solo traveler, and where I could get meals without seafood. I'd also want it to be in a place where I could spend the day on easy-moderate hikes or walks.

• I will be traveling alone. As of now, I'm planning to stay in Hakone for 1-2 weeknights. Will I have a problem booking a room in a ryokan for just one person?

Yes. Many ryokan don't accept single travelers. Or, they might accept a solo travler sometimes, but not every day. You might find the perfect ryokan but have to search day-by-day to find availability. Ryokan stays typically include meals, so they charge per person.

As a rule, you should expect lodging prices to depend on the number of guests in the room. You should search for one guest but if you find no availability, search again for two guests to see if there is any availability at all. If you find availability for two guests on a date you want, you can then try again later to see if the room becomes available to a solo guest.

• I don't like any fish/seafood and I'm just generally picky about meat, so I'm thinking a non-seafood or vegetarian meal option would work best for me. Is that possible at ryokans? I know you generally can't customize meals in Japan. (I don't mind dishes cooked with fish sauce or other seafood products, as long as it doesn't taste fishy to me, which I realize is pretty subjective.)

This limits your options much more than traveling solo. Japanese eat rice & seafood, and ... other stuff. Ryokan breakfast would typically include rice and a piece of fish. Thd fish isn't going to be fishy smelling.

Some places do offer lodging only with no meals.

Some places do have multiple meal options which might include veg* optons. You can find ryokan using OTA webistes (e.g. japanican or agoda), but you should go to the ryokan's website to see the meal options. And book with their website to get the meal options you want.

I could have booked a ryokan at an OTA but all it offered was the standard kaiseki meal but the ryokan website had several options. So, I booked directly with the ryokan using their Japanese-only website (using a translator plug-in)

• Ideally I'd love to stay at a ryokan with a gorgeous outdoor onsen that feels like you're in the middle of a garden. My budget isn't set in stone, but I'm thinking $700 for a two-night stay, although I can cut it down to just one night (keeping the $700 budget) if I need to. The onsen doesn't need to be in my room, but ideally it would have times available to book privately. Not a deal-breaker if it doesn't though; I'd prefer a beautiful single-sex shared onsen to a private tub in what looks like a normal bathroom.

Your $350/night budget for one person is a healthy one.

Any specific ryokan recommendations that meet the criteria above?
Yes, but not near where you are staying.

Many/most onsen ryokan make their bath(s) available to non-guests during certain hours. In Hakone I stayed in a minshuku but visited a ryokan to use their onsen.

I'm also open to recommendations for an onsen town other than Hakone. It would need to be somewhere I could get to fairly easily between Tokyo and Kyoto,
Atami Onsen, but I just went to a ryokan for a soak and did not stay.

We are planning to visit Japan this October. We are staying in Hakone at Byakudan, each room has their own outdoor hot springs bath. Traditional meals (breakfast and dinner) are served in your room. They also have larger public bath with views of garden and distant mountains. This is our first time staying at a ryokan but excited. You may want to check this place out.