Train to KIX
I did the train a couple of weeks ago. When getting to Shin-Osaka there were two options.
1. Take the TR to Osaka station (I should have done this). Leaving the station the hotel is across the street.
2. Take the subway Midosuji (Red Line) (M13) to Hommachi stop(M18) cross to the Yotsubashi Line(Blue)(Y13) to Nishi-Umeda(Y11) station. The hotel is above you. I think exit 6.
I took choice 2. It wasn't terrible, but quite a walk between the red and blue lines while dragging bags. Next time I will take the TR.
My advice. While your at the Shin-Osaka, go to the bottom floor and look for the noodle shop on one end with a Red Box sign. Good noodles, enjoy lunch and a beer, then continue your journey.
Oh and by the way if you are new to the train thing in Japan. You'll get basically 2 boarding passes and a ticket from NRT-KIX. Don't lose any of them, especially the ticket. That one cost me another 88 bucks. They won't reprint it for you even with a receipt.
Good Luck!