transport for tokyo through to osaka

Hi, I will be in Japan in Feb 2019 for a 6 day trip. First, flying to Tokyo Narita, heading to Karuizawa for the first day. Then, second day, head to Takayama and Shirakawa-go, staying in Takayama. Third day, head back to Kyoto for 2 days and last but not least, Osaka for another two days.

My question is whether Hokuriku Arch Pass or JR national pass is more worthwhile. I understand for the leg from Takayama and Shirakawa-go would be out of that boundary.

If you could provide me with some opinion or options, that would be great.

Welcome to my country!

I will be in Japan in Feb 2019 for a 6 day trip

You will stay until Day 7, won't you?
(I understand you will be In Kyoto until Day 5 then in Osaka until Day 7.)

First, flying to Tokyo Narita, heading to Karuizawa for the first day.

Actually Narita Airport [NRT] is outside Tokyo Metropolis.
When in the day is your flight arrival estimated?

Have you checked whether you can choose Haneda Airport [HND]?
Haneda Airport is in Tokyo's urban area and a little closer to Karuizawa.

Then, second day, head to Takayama and Shirakawa-go, staying in Takayama.

You can travel
from Karuizawa by JR Hokuriku Shinkansen
to Toyama / Kanazawa
then by non-JR expressway bus
to Shirakawa-go (1700 yen / 1850 yen)
and further to Takayama (2470 yen).
The total amount of the bus fares is 4170 yen / 4320 yen.

Would you visit also Kanazawa?

Third day, head back to Kyoto for 2 days

You won't head "back" to Kyoto,
because you will go not via Kyoto for Takayama.

and last but not least, Osaka for another two days.

So, you will fly out from Kansai Airport [KIX].

My question is whether Hokuriku Arch Pass or JR national pass is more worthwhile.

Japan Rail Pass sounds better for your itinerary (as is planned now).

Japan Rail Pass is valid
from Takayama via Gifu to Nagoya
(by JR Takayama and Tokaido Lines: Limited Express)
then from Nagoya via Maibara to Kyoto
(by JR Tokaido Shinkansen: except "Nozomi").

When you take the Limited Express train departing at 8:00,
which is to arrive to Nagoya at 10:34,
you will catch the 10:46 Shinkansen Kodama,
which is to arrive to Kyoto at 11:34.

Total amount: About 33400 yen.
: 29110 yen (#1) paid for a 7-day Ordinary-Class JRP
(when it is purchased outside Japan).
: 4320 yen paid as bus fares (when you visit also Kanazawa).

There are some options
which are a little slower but a little less expensive.
I have not found the one
which is much more cost-saving
and not much more time-consuming.

Hokuriku Arch Pass
does not cover JR routes from Takayama via Gifu to Maibara.
The amount of the additional cost is,
compared to the gap in the prices (about 5000 yen)
between the two passes,
estimated as about the same or higher by a few thousand yen.

Total amount: About 33800 yen to 36700 yen.
: 24000 yen (#1) paid for a Hokuriku Arch Pass
(when it is purchased outside Japan).
: 5510 yen to 8340 yen (#2) paid as JR fees
to travel from Takayama until Maibara.
: 4320 yen paid as bus fares (when you visit also Kanazawa).

(#1)
You pay for the pass in your currency.
Here a change in the foreign exchange rate is not considered.

(#2)
The additional cost to travel until Maibara is
7560 to 8340 yen
when you travel beyond Nagoya by Shinkansen
(as is mentioned above), or
5510 to 6030 yen
when you travel wholly by non-Shinkansen lines,
taking the only direct Limited Express train for Kyoto,
which is to depart at 15:38 from Takayama
and to arrive at 19:17 to Kyoto.
Possible slower options are ignored here.