There are no bullet trains from Kansai Airport to Shin Osaka. While I can hear your disappointment as I write this, there are other options that are just as comfortable.
If money is not an issue, the fastest and most relaxing way is to use Japan Rail (JR)’s Haruka limited express. It takes 51 Minutes with no transfers for the 60.8 km trip, and costs ¥ 2,850 (consisting of the ticket fare of ¥ 1,360 and the seat fee of ¥ 1,490). You can use the Japan Rail Pass as well.
Haruka services generally depart every 30 minutes during peak hour, and hourly during the off peak hours. Train timetables are located at Hyperdia.
The first train from the Airport is 6:30am and the last train is at 10:16pm, so if you miss it, take the slower normal express, which takes 84 Minutes with one transfer at Osaka Station.
If you don’t hold the Japan Rail Pass, the Haruka can still be value for money. Overseas Tourists can buy a 1 day JR West Kansai pass. It is only 2,200 yen with the early bird discount and gives you unlimited access on the Haruka plus all other JR trains within Kansai area, including Nara, Kyoto and Himeji. You cannot ride bullet trains with the JR West Kansai Pass. You don’t have to pre-purchase the JR West Kansai pass before you arrive in Japan, you can get it from the ticket office at JR Kansai Airport Railway Station, and it only takes five minutes once you show your passport.
Just remember, the Haruka does not stop at Osaka station. If you want to go to Osaka, you have to change at Tennoji or Shin Osaka for a local train. It can be slightly quicker to change for the subway at Tennoji, but it will cost more, and it is an effort to go from Tennoji JR to Tennoji Subway, through the busy thoroughfare with your luggage. I don't recommend it. Even if you are in a hurry, you might get lost. It may be easier to change to a JR local train instead at Tennoji.
If you prefer the Nankai Rapit service, you can also get to Shin Osaka in a little over 60 minutes, with a change at Shin-Imamiya or Namba.
Alternatively on the local express, it takes 84 Minutes with one Transfer at Osaka Station. It can be slightly quicker to change for the subway at Tennoji, but it will cost more, and it is an effort to go from Tennoji JR to Tennoji Subway, through the busy thoroughfare with your luggage. I don't recommend it. Even if you are in a hurry, you might get lost.
For 2019 Rugby World Cup fans heading to Hanazono rugby stadium, you can take either the JR train to Tennoji or the Nankai train to Shin-Imamiya, for two connecting trains to Higashi Hanazono station, a total journey time of 1 hour and 15 minutes.