Does the America the beautiful pass cover natural bridges national monument?

yes

Covers all Parks, Monuments, Rivers, Seashores, Battleifelds. Some places it doesn't cover. Like parking at Mount Rushmore, for example. I think ,at Carlsbad Cavern, it covers your entrance to the cave, but does not cover or apply any discount to one of the tours.

But to most of the major parks cost $25 to enter. Some are free. Like Capitol Reef and Smoky Mountains are both free. If you arrive at some parks early morning, there is nobody at the booth and you enter free(Arches and Grand Canyon). You may be asked to pay as you leave, but most likely you won't be. I've bought a pass myself the past 5 years.

It covers most federal lands.

It does not cover state, county or local parks. You can ask for a discount and some may give it.

I bought my lifetime pass the first time I went thru Zion in 2007.

Capitol Reef and Smokey Mountains don't charge because the highway that is used for normal transportation goes right thru the park.

I don't know the layout for GSNP but I asked about that at Capitol Reef and they told me it would cost more to set up pay stations than it would be worth. Several of the most popular hikes are directly off the highway going thru the park.

An interesting article about NPS fees.
It says that there are 265 units that dont' charge fees. I don't think I paid anything at Guadalupe, Redwoods, Death Valley, Hot Springs.

http://www.nationalparkstraveler.com...lebration22900

As to Capitol Reef. They do ask if you drive the southern Scenic Drive to pay $5 at the visitor center. So not completely free.

The pass does not cover Indian Tribal lands either such as Antelope canyons in Page, or Monument Valley.