When a Ring says something like "Skirmish: Add a burden to wear The One Ring until the regroup phase," that's a special ability as follows:
Costs: Add a burden.
Effect: Wear The One Ring until the regroup phase.
If you put on the Ring through another means, you don't have to use the ability, so you don't have to pay the costs.
It's like if you're at a sanctuary with
Boromir, DoMT, and heal Boromir using the normal sanctuary healing (rather than his ability), you don't need to add
![2 [2]](https://lotrtcgdb.com/forums/Smileys/classic/2.png)
because you didn't use his ability, you healed him by other means. Similarly, at
Narchost, you didn't put on the Ring using the Ring's ability, so you don't need to pay the costs for that ability.