Check the Tempozan (Kaiyukan) bus departure before buying a train ticket. A service listed under the wider Nanko-Tempozan-USJ route does not always stop there.
Sumiyoshitaisha Station is served only by Local trains, so change before Namba. Coming in on an Airport Express, Sakai is the practical place to switch.
Shin-Imamiya, Ebisucho or Dobutsuen-mae, not Namba. Rapi:t and Airport Express both stop at Shin-Imamiya, and Nankai lists Tsutenkaku as 10 minutes on foot.
Umeda Sky Building belongs to the Osaka Station side, not Namba. JR from the airport reaches Osaka Station in about 65 minutes, then you walk past Grand Front.