First timers - Tokyo and Kyoto or just Tokyo

There is plenty to do and see and experience in Tokyo, and if you include an overnight trip to Nikko and/or Hakone, it is not a problem at all to spend your whole 9 days in Tokyo without getting bored. Having said that, Kyoto is also an amazing city, and I would probably do 5 nights Tokyo (but then no day trips) and 4 nights Kyoto.

A couple of comments when I consider your interests:
- Ramen: you can get amazing ramen all over Japan. One place that you might consider if you do include Kyoto, is the instant rhttps://www.cupnoodles-museum.jp/en/osaka_ikeda/amen museum in Osaka (its easy to do a day trip from Kyoto to Osaka). If you like spicy food, have a look at this vlog https://www.youtube.com/watch?v=jrMsWztL1n8
- Nature: if you are thinking mountains, small villages surrounded by beautiful nature etc. - you obviously wont get that if you stick to Tokyo and/or Kyoto. Having said that, there are several beautiful parks in Tokyo, and in Kyoto several temples have wonderful gardens, and you can some really nice bamboo groves - for example at Fushimi Inari shrine in Kyoto.

Are those 9 full days or does it include arrival/departure days?

Is there enough for 9 days in and around Tokyo if we do an overnight to Hakone or Nikko?

Certainly, and particularly with your interests. IMO, just for the highlights in Tokyo you'd need around 3 full days. You probably want a couple more days for Tokyo. Two days for Nikko for nature, onsen and some of the best temples/shrines. Hakone is also good for a day. And Kamakura/Yokohama could fill another day.

We don’t need JR pass for that right?

No, definitely not. If you only stay in and around (Nikko, Kamakura, Hakone, Fuji...) Tokyo, a JR pass would be a huge waste of money. You could consider local rail passes like the JR Tokyo Wide Pass, the Tobu Nikko Pass or the Hakone Free Pass.

Tokyo + Kyoto is of course a very reasonable alternative, but would basically limit you to the highlights only. Also, it could be slightly more tiring and you'd pay considerably more on transport.

Do we need a JR pass if we choose this option? I looked up trains on hyperdia and it looks like we’ll break even buying JR pass just getting from Tokyo - Kyoto - Tokyo.

The JR pass should be a bit more expensive than just Tokyo-Kyoto-Tokyo. In addition you'd need at least one medium distance trip (e.g. Tokyo-Nikko-Tokyo) to get even. Probably you don't have time for this and buying regular tickets would be the way to go (which have the advantage that you can take the faster Nozomi shinkansen).

• Nightlife and/or good drinking: not at all a problem. Shinjuku is one (if not the best) area to stay in Tokyo for nightlife, it is very lively and you can find pretty much everything, from lively half open-air Izikaya stalls in Piss Alley, to the "crazy" show at Robot. In Kyoto you can also find nice little sake bars, or places to have a couple of beers and people watch.
• Manga, anime, cute stuff: yep, Tokyo is totally the place for that. Akihabara electric town and Harajuku are especially "good" for that. The Ghibli museum (I haven't visited yet) should also be amazing, but it could be quite difficult to get tickets in such short notice.

Temples: Kyoto has many many many many temples and shrines. I love temples and I stayed in Kyoto for 7 days, and could only visit a very small amount of the many temples there.
Ryokan: staying in a Ryokan is indeed something I would do for at least 1 night (2 would be better). Its not cheap, but its a really great experience. Do choose a Ryokan that offers half board, as food is one of the main highlights of staying in a Ryokan.
Onsen: a lot of Ryokan have an onsen, and again - a great experience. So if possible, choose a Ryokan with an onsen, and as you have a tattoo, choose one with a private onsen. I dont think that it is possible to visit a "normal" onsen if you have a tattoo (unless its very small and you can cover it up under a plaster or a bandage).

Agree with above.There is plenty to fill your time in Tokyo and around.

OTOH Kyoto is my favourite city in Japan,and you have enough time to see both,at least for an introduction.

No need to go to Kyoto. Kanto(Tokyo and area around) has more than enough to do for 9 days and given your interests I dont really see the need to trek down to Kyoto to overload on temples.

The Ramen Museum/village in Yokohama is something you might want to add to your itinerary if you stay in and around Tokyo. There are dozens of varieties, availible in small portions so you can sample many. It could make for a good evening with a stroll along the renovated water front after a day soaking up Kamakura's temples.

Are those 9 full days or does it include arrival/departure days?

That includes arrival and departure days. We actually arrive at HND at 3:25pm on 11/14 and depart at 7:45pm on 11/22.

The JR pass should be a bit more expensive than just Tokyo-Kyoto-Tokyo. In addition you'd need at least one medium distance trip (e.g. Tokyo-Nikko-Tokyo) to get even. Probably you don't have time for this and buying regular tickets would be the way to go (which have the advantage that you can take the faster Nozomi shinkansen).

I'm assuming Hyperdia results are one way right? I generically put in Tokyo to Kyoto and came up with 13,910 (reserved) 13,080 (unreserved) on SHINKANSEN Nozomi. Which, roundtrip is about $230 USD. I guess its a bit less since the JR Pass is $242, not a huge cost savings but still worth it to buy individual passes if we choose to visit Kyoto?

The JR pass should be a bit more expensive than just Tokyo-Kyoto-Tokyo. In addition you'd need at least one medium distance trip (e.g. Tokyo-Nikko-Tokyo) to get even. Probably you don't have time for this and buying regular tickets would be the way to go (which have the advantage that you can take the faster Nozomi shinkansen).

I'm assuming Hyperdia results are one way right? I generically put in Tokyo to Kyoto and came up with 13,910 (reserved) 13,080 (unreserved) on SHINKANSEN Nozomi. Which, roundtrip is about $230 USD. I guess its a bit less since the JR Pass is $242, not a huge cost savings but still worth it to buy individual passes if we choose to visit Kyoto?

The JR pass cannot be used on the Nozomi, you have to take the slower Hakata or Kodama services if you use the pass. The Kodamas are seriously slow.

The pass is virtually useless when you are in Tokyo or in Kyoto, as they do not work on buses, subways, or most local train lines. You might get some value out of a day trip to Nara, but that can be done for $4-5, so not a lot of value.

If you can use it to get to or from the airport, and that airport is either KIX or NRT you can save $10-15

Remember, JR is selling the JR pass to MAKE money. Most people who buy it end up paying more than if they had just bought individual tickets.

I think Tokyo is cool city, but Kyoto is the must visit city in Japan, there are more beautiful landscape there, you can enjoy the incredible tranquility and japanese culture in Kyoto only. Just take 新幹線，few hours you will arrive kyoto.