Harajuku is the cleaner JR arrival, Meiji-jingumae the cleaner Metro one. Official access times reach the entrance only - the main shrine is 10 minutes further.
JR Universal City Station is the reliable anchor, about 75 to 85 minutes from KIX. The airport bus has several USJ-area stops, and your hotel name decides which.
HARUKA reaches Tennoji in about 30 minutes, the Rapid in 45. The dedicated airport bus to Abeno Harukas is currently suspended, and Tennoji is not the entrance.
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.