Worth of JR pass

Hi,

Should we get JR pass?
1st day: Arrive in Nagoya
2nd day: Travel to Tokyo (for 3 days)
4th day: Travel to Osaka (for 4 days), hotel in Osaka but traveling back and forth to Kyoto, Nara, etc.
8th day: Travel to Fukuoka (for 2 days) and exit here in Fukuoka airport

Other questions:
- how many hours from Osaka to Fukuoka - do you have any hotel recommendation (or just the location) in Osaka near JR stations that is a strategic location (near shin-osaka station and near sightseeing place)

Thanks in advance!