JR Kyushu Pass - Questions

I have two questions relating to the JR Kyushu Pass:

1) Are there JR or other lines within Fukuoka City where I can use this pass? For instance, for travel between Hakata and Tenjin stations, or from Fukuoka Airport to Hakata station.

2) I understand that the pass can be purchased at TISCO Travel Information Center at Fukuoka International Airport. What time does the center open on Saturdays?

Thank you very much for any advice.

Are there JR or other lines within Fukuoka City where I can use this pass?

Yes, all the JR lines in Fukuoka can be used (except Sanyo Shinkansen). They are the green colored lines on the following map:
http://yokanavi.com/access_info_eg

For instance, for travel between Hakata and Tenjin stations

No, that is not a JR line. It is a subway line.

or from Fukuoka Airport to Hakata station.

No, that is also a subway line.

I understand that the pass can be purchased at TISCO Travel Information Center at Fukuoka International Airport. What time does the center open on Saturdays?

I believe the opening hours are 8:00 to 17:00 on every day of the year:
http://www.tiscojapan.com/exchange/counter/

As said, yes, but not much to places that you'd really want to go to. Hakozaki has a good shrine that is very historical (it was burned down in the first Mongol invasion).
https://www.youtube.com/watch?v=sCuNuGapQxw
About 20 minutes outside the city though in Sasaguri in the Nanzoin Temple - one of the best places you'll ever see in the country.
https://www.youtube.com/watch?v=lT4VQFmHFyM
There are also trains, though infrequent, going to Uminonakamichi.
You would probably do better to get around using a Fukuoka Tourist City Pass, or you can cover a lot of places with just a subway day pass.
http://yokanavi.com/access_tourist_eg
There are some very nice places to see though. Look through a site like
http://www.thejapanfaq.com/fukpic.html
to find something you like.