Do I need a JRP to ride Shinkasen

First, I am aware that there are alternate train routes that I could take that are not using the Nozomi trains.

My question is simple. Do I require a JRP to buy tickets for the shinkasen?

I only need 1 way tickets from Tokyo to Fukoaka and only on 1 day. I will not need passes for another week and a JRP is awfully expensive for a single day use.

If you use JR only to travel once from Tokyo / Shinagawa through to Hakata (in Fukuoka), a Japan Rail Pass does not pay off. The regular JR cost is up to 23150 yen per adult (including an additional charge on Shinkansen) as long as you choose Ordinary Class and take the Shinkansen train(s) on a same one day.

Traveling with regular JR tickets, you can take Shinkansen Nozomi and Mizuho trains, which you cannot take with a Japan Rail Pass.
Usually you get issued plural JR tickets for your trip by Shinkansen: one ticket indicating your whole route, and one or a few tickets for you to use Shinkansen.
Maybe you get issued one JR ticket which works as the both if you take only one Shinkansen train or if you do not use a reserved seat.

Do I require a JRP to buy tickets for the shinkasen?

Of course not... Otherwise, the great many people who don't qualify for it would be unable to take the Shinkansen.

If you don't mind 5 hours on a train, you can take the Nozomi straight through for 22,950 yen.

Also, if the only trip you want to do is Tokyo to Fukuoka, then flying is both a lot faster and a lot cheaper than the train (even if you fly with JAL or ANA and not a low-cost carrier).

Agree with Fiars, Air is better and cheaper if you just going Tokyo to Fukuoka

Unless you're taking the flight at the last minute it will always be cheaper to take the plane for distances that cost more than 20k yen. It's also much faster.

Unless you need to go to places with no airport taking the shinkansen for very long distances isn't worth it, if you're ready to spend so much time travelling in the first place, a night bus will be cheaper (depending on period can be more or less expensive than plane) and you get something as convenient as the train for boarding (you seriously just need to show up and say your name and reservation number and they didn't even ask for it last time for me).

Why not just fly? You can take a low cost carrier like Jetstar for as low as 6200 yen one way - a nice bargain. See:
http://www.jetstar.com/jp/en/home
You arrive in only about 2 hours.
Then you can use that extra time for more exploring!
A week in Fukuoka is a large chunk of time, but there are many interesting places to see if you want to - and they're not overrun by crowds of tourists either.
For a list of several good places to go and cheaper places to stay, try looking at:
http://www.thejapanfaq.com/fukpic.html
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...
While there, look into a Fukuoka Tourist City Pass at:
http://yokanavi.com/access_tourist_eg
There are many other areas close by for day trips as well - Kurume has its 62m great buddha and temple:
https://www.youtube.com/watch?v=G1SBtmqXteE
There are also the canal boats in Yanagawa, and Dazaifu is a quaint historical area with its famous tenmangu shrine:
https://www.youtube.com/watch?v=BwxGPZxPlNw
And if you want to see more recent history, close by is the kamikaze museum in Tachiarai:
https://www.youtube.com/watch?v=_fBxvz48E5w
Plus if you want to stretch out a bit more, Kitakyushu is not far and has a nice castle in Kokura, historic Moji wharf area and Sarakurayama, one of the best and nearly unknown mountaintop views in Japan:
https://www.youtube.com/watch?v=Hl1pCW9Ibbk
But don't think you're stuck taking the train...these days there are a lot more options where you don't have to pay through the nose anymore.

Thank you for the helpful responses. Everything I found stated you couldn't get shinkasen tickets online, and mentioned that the JRP didn't work for shinkasen, but not what to do about getting shinkasen tickets in advance.

Thank you for the details for plane information and sights in Fukuoka. I looked into planes and everything I found was $400, even ANA flights. That seems significantly more reasonable.

- japan-guide.com: Discount Air Tickets
http://www.japan-guide.com/e/e2364.html

Peach Aviation and Jetstar are so-called low-cost carriers which have domestic Fukuoka flights.
They are using not Tokyo International Airport [HND, Haneda Airport] but Narita International Airport [NRT], which is actually outside Tokyo.
Peach Aviation is using NRT Terminal 1 (South Wing) while Jetstar is using NRT Terminal 2.

When you head to NRT by rail in order to catch a flight, please note that Narita Station and Keisei Narita Station are outside the airport. Trains running to NRT make the second last stop in Terminal 2 before arriving to Terminal 1.

- japan-guide.com: Narita Airport
http://www.japan-guide.com/e/e2027.html