Do I need upgraded rings for the achievement?

Yes, you need all the rings versions to get this achievement.

That means, you can't get it until NG++.

From Achievements:

Master of Rings Acquire all 107 rings. (NG++ Required)

Yes you have to collect every ring including their upgraded counterparts as stated on the Dark Souls 3 rings wiki page:

To obtain the Trophy/Achievement for collecting every ring, you must collect all of the different +X variations available, making a total of 107 rings.