Obito's Kaleidoscope Sharingan has a power that allows him to become invulnerable to attacks, as if he was a "ghost"; Kamui:

I was wondering if the Susanoo carried a similar ability.

The answer seems to be yes. In the Susanoo's case, it seems to be adjusting it's barrier to allow objects through it, rather than sending things to a different plane of existence. To some extent this is superior to Obito's Kamui, since parts of the Susanoo can remain tangible while others are intangible, and the user can still attack while protected by Susanoo.

While Obito's Kamui restricts his attacking potential. Although I suspect Susanoo cannot make the user themselves intangible, which could explain why they don't use this ability more often.

I guess the guy that wields "spirit weapons" can become a spirit himself.