Should I buy a JR pass? (Osaka - Tokyo)

I travelling Japan together with my fiancee on Oct for a trip of 11 day 10 night. We will be touching down in Osaka and stay for 5 nights. Meanwhile, the 5 days we will also travel to Kyoto and Nara before heading to Tokyo on the 6th day.

Do I need to buy a JR pass or there are other better option on this?

If you are departing from Tokyo, then no need for JR Pass. 1 way shinkansen ticket to Tokyo will do, within Osaka, Kyoto & Nara just use the IC card.

I will be flying to Osaka first and stay for 5 days (Including travel to Kyoto and Nara). On the 6th day I will be travelling to Tokyo (Staying another 5 days) and fly off from Tokyo.

You may want to consider flight from Itami to Haneda which which is cheaper than Shinkansen.

Plane might be cheaper, but your time also has some price!

Shinkansen leaves from central Osaka and arrives in central Tokyo. Connects directly to loops in either city.

Airplane. You drive to the airport. You have to be there 1-2h early to clear. Then you arrive in TKY and again you have to get from airport to central TKY.

Now, what is cheaper and faster?