Transportation from Osaka to Fukuoka

Hi there,
We are arriving at Osaka KIX Airport on 18th Oct 14 (Sun) via MH 88 @ ETA 7.40am.
From KIX, we are heading for Fukuoka.
We notice that is the cheapest way is by LCC flight - Peach Airline/Jetstar.
However the connecting flights is either early/later.

1) Peach Airline
MM153 ETD 10.25am
MM157 ETD 4.35pm
2) Jetstar
GK555 ETD 2.20pm

Our questions are:
1) Is enough time for us to catch MM153 as our arrival time is 7.40am (via MH88)?
2) Do you recommend us to travel by Shinkansen?/train as the price is expensive.

We are actually joining a local group tour operator on 19th Oct 14 (Sun) & to meet at 8.30am at Fukuoka Airport. So we are planning to either stay 1 night in Fukuoka or spend 1 day in Kansai area and catch the earliest PEACH flight MM151 ETD 7.10am on 19th Oct 14.

What is your recommendation what we can do in Osaka for 1 day travel?

Thank you for your assistance and advice.

Best regards,

Nicole

Barring a disaster, you should be able to easily catch the 10:25 flight.