Japan first time: Itinerary and questions

Hi everyone,

I will be travelling to Japan for the first time and many years of wishing to do it. I will spend 5 days in Tokyo and 3 in Kyoto (including Nara half a day).

I have some questions please:
-What is the best and fastes way (no taxi) to reach Shinjuku from Narita airport? I can't decide beteween NEX and Limousine Bus. I've read that the limousine Bus is cheaper but takes longer? I'll be staying at the Nishitestsu Inn Shinjuku. In both cases, where can I get the tikets from?

-Regarding the hotel, can the handy phone that the Nishitestsu Inn Shinjuku provides, be used as a pocket wi-fi, or do I still need to buy a pocket wi-fi? I've sent an email to the hotel but got no reply.

-I do not have a JRP, to go from Tokyo to Kyoto, is the Nozomi train a good solution?

-What would be the best option to go from Kyoto to Nara and back? I do not have a JRP.

Thank you

I can't decide beteween NEX and Limousine Bus. I've read that the limousine Bus is cheaper but takes longer?

The NEX can be considerably cheaper if you use the NEX Round Trip Ticket:
https://www.japan-guide.com/e/e2359_nex.html

Also check the available drop-off stations of the bus. They may have one that is closer to your hotel.

Apart from that I think it is a matter of personal preference. Some people prefer trains others - for some reasons beyond me - prefer buses ;) I prefer trains because you have more space, better views and timetables are followed quite reliably.

In both cases, where can I get the tikets from?

At the respective companies' ticket counters or machines. The bus ones are in the arrival lobby, the train ones are in the basement next to the station ticket gates.

Regarding the hotel, can the handy phone that the Nishitestsu Inn Shinjuku provides, be used as a pocket wi-fi

I think you can get the answer only from the hotel directly. It seems to depend on the hotel whether they allow the handy phones to be used for tethering or not.

I do not have a JRP, to go from Tokyo to Kyoto, is the Nozomi train a good solution?

Yes, it is the fastest and most convenient solution. But there are various alternatives, and it depends on the individual traveler what is best for them:
https://www.japan-guide.com/e/e2363.html

What would be the best option to go from Kyoto to Nara and back? I do not have a JRP.

For comfort I recommend the direct Kintetsu limited express and for a lower budget I recommend regular Kintetsu trains. All the options are explained here:
https://www.japan-guide.com/e/e4107.html

For the N'Ex vs bus question, a lot depends on where you ultimately plan to end up. The bus is expensive, but if you are staying at or very near where the buses go (like the big name tourist hotels) it can be virtually door to door service. And after flying around the world, you will likely arrive dead tired. The N'Ex can be quick though, and if you are staying very close to the station, can also serve you well. Or you can grab a taxi at Shinjuku Station if you want.
If you are not getting the JR Rail Pass, then why not take the Nozomi trains? You'll get there 30 minutes faster than the Hikari. If you want to save money, and you are there less than a week, then one option is the JR Flex Rail Ticket
https://www.jrtours.co.jp/en/
And yes, you can take the Nozomi trains with it.
For Kyoto to Nara, you can take JR or Kintetsu - the latter has its Nara station closer to the action if you are going to Nara Park, where most people go.
https://goo.gl/LfvWnL

You also have the option if you still have the energy to go see Osaka at night after a day of sightseeing in Kyoto. Osaka really comes alive at night and there are some good places to see - Dotonbori, Shinsekai, the Umeda SKy Bldg, Abeno Harukas Bldg, Cosmo Tower, etc.

Take the N'ex. It's easy, fast, not expensive and goes to Shinjuku. The last thing you want after a flight is hassle.

The Handy Phone is very good and you can take it with you while out & about. I had one at a hotel on my last trip & it proved a lifesaver (when something out of my control came up). I didnt dig into the settings to see if I could tether to it but I expect it to be locked down.