Yeah, the crew was given a pill called (I believe) Provigil which is primarily Modafinil, an anti-sleeping pill that basically takes over all the receptors in your brain that respond to hormones and chemical build-ups that cause drowsiness and prevents your brain from feeling tired. It's not healthy, really, but they did it. I believe he said they could go from somewhere in the Mid-West or Southern states where they have a giant airbase that primarily houses and repairs B-52s for the whole Air Force all the way to Saudi Arabia on one tank of fuel and would do a series of mid-air refuels in the area during the sortie before coming back home. And no, they weren't allowed to sleep, not even in shifts. He did land a couple times in a major Saudi airport when no air tankers were available or they needed to repair any damage (never from being shot at, but when you go up to the altitudes these things fly at it's like an SR-71 where the metal expands from the low pressure and shit just breaks now and then) and that's how I got my first "Rolex", which actually worked pretty well and was self-winding but completely fake despite looking identical to a popular model at the time.

Shifts would have been difficult to manage as the B-52 was designed as a long-range bomber and would take a week to go out and come back from missions, the main reason it was used was the dearth of friendly air force bases in the area, the ones which were available were being used by fighters and smaller recon jets and bombers and most couldn't accommodate the B-52's massive size even if they were empty because the runways weren't wide enough. If you're talking about the pilot/copilot taking shifts, that wasn't allowed or possible. The pilot would typically be the primary flier while the copilot would function as a navigator and bomb controller, sleeping would just make things difficult if the shit hit the fan and AA was locking on or the President called to authorize a nuclear strike.

Neat fact most people don't know about the B-52: it was one of the first bombers to include an ejection seat, and the ejection seat shoots downwards due to the fact that 90% of the time these jets are at altitudes which would kill a pilot very quickly if they were shot upwards into even thinner air. The gunner seat didn't have an ejector but they did have plans to include one that would basically turn the entire tail section into a glider, but this was never implemented successfully or even tested I think. The more you know!