Hi Ichi,
On 4/12, can JR get us from Kagoshima/Ibusuki to Narita airport in time to check-in and board for our 8PM flight?
In order to board the 8pm flight, you should aim to be in Narita by 6pm. While you can travel from Ibusuki on the same day this will mean a very early start getting the train at [7am and arriving at 5:30pm]. As this trip will be followed by a flight, you may find it hard going. Here are some tips:
Showers and day rooms at Narita
At Narita you can rent a shower room or small bedroom by the 30 mins period. I really recommend you use these to refresh yourself before your flight - they make a huge difference, are very clean and well organised, and cost around ¥500 per session.
Consider travelling up Japan a little the day before
You'll have an easier time if you aim to get the train from Kagoshima, or perhaps try to travel up the country a little the day before. How about overnighting in Fukuoka - there is great cuisine in the Hakata district and it is a bustling city. Alternatively, you could think about a day and night in Hiroshima/Miyajima which will break the journey back up to Tokyo nicely.
Hope this helps!