I believe you have two options for this.
Option One - Shinkansen
Firstly, take the Narita Express to Tokyo Station (3020 yen).
You should easily be able to find tickets at Narita airport for this. But more information can be found here: http://www.jreast.co.jp/e/nex/
Then, once in Tokyo Station you can get the Shinkansen to Takasaki.
Make sure you are on the Shinkansen Toki on the Joetsu line.
At Takasaki, exit the Shinkansen and head towards the JR lines and get on JR Joetsu Line which is bound for Minakami.
Tokyo to Minakami should cost you 5490 yen.
Total Time: Just under 4 hours
Total Cost: 8510 yen
Option 2 - No Shinkansen
Take the Narita Express to Tokyo (3020 yen)
Then, take the JR Keihin-Tohoku line to Ueno.
At Ueno, change trains to the Takasaki line, and ride it all the way to Takasaki.
Then at Takasaki, change trains to JR Joetsu Line which is bound for Minakami.
Cost should be 3020 yen.
Total Time: Just under 5 hours
Total Cost: 6030
I tried looking for a bus, but I haven't found a semi-direct one yet.
As for taxi, I'm afriad the trip is a 3 hour drive, and falls outside of the fixed fare taxi's available at Narita aiport. As an estimation, you are looking at over 50,000 yen for a taxi there.
You can look up the aiport's taxi details here: https://www.narita-airport.jp/en/access/taxi
Hope this answers your question.