Your question is better suited for Tripadvisor (I am the Osaka expert under a different name there).
Anyway, if you opt for Kyoto, get the 1-day Kansai Area Pass. The Haruka train takes roughly 75 minutes from KIX to Kyoto. You can visit a couple of places. Consider Kiyomizu-dera and Higashiyama to start, plus a couple shrines in the area nearby. Finish at Gion, head back to Kyoto station and onto KIX.
If you go for Osaka, there might not be too much going on so early. You could start your day taking the train to Tennoji. Atop Tennoji station, the Abeno Harukas observation deck offers a stunning view. The Kintetsu Department store just beneath is excellent. It also has a lot of food choices but most only start opening for lunch. Before or after lunch, take the train to Namba and roam Namaba and Dotonbori just to people-watch and take in the atmosphere. When you are ready to head back to KIX, take the Nankai train (Airport Express commuter style train or the more expensive Rapit) back to KIX. Both trains take around 35-45 minutes. Avoid the local train that stops much more frequently.
Jourdan is a good train site but I tend to use Hyperdia:
http://www.hyperdia.com