After doing some research, I opted not to do it — yet. Organizing to hike Paria Canyon was easier.
The best of several options, I thought, was to hike in to Rainbow Bridge, then catch a tourist boat back to Wahweap marina near Page, Arizona. (about US$55)
If you can find transport to the trailhead early in the morning this one-day, light-weight hike is feasible. The hike in is via either of 2 foot trails: from Navajo Mountain Trading Post, 14mi (26km), or from abandoned Rainbow Lodge, 13mi (24km). Details on GORP.
A longer alternative is the Rainbow Bridge Trail: a 28mi circuit from and back to Rainbow Bridge. You would boat both ways and spend some quality time close to the spiritual monument.
Any hiking longer than the tourist 90min tour requires you get a Navajo Nation backcountry permit.
While at 275ft (84m) long and 290ft (88m) high it’s neither the highest nor the longest arch / bridge in the world, Rainbow is certainly the most famous.
I’d love to see it … someday.
Rainbow Bridge National Monument – Wikipedia
besthike.com 
Nice Arch route!
We visited Arches & Canyonlands first time last year. Hiked and learned about canyoneering. Really was impressed by this area.
DSD
“Summit Stones & Adventure Musings”