In this scene:
At 3:03, General Grievous breaks the bridge glass with an electrostaff and jumps out, and Anakin and Obi-Wan hold onto control panels to not be sucked out by the loss in pressure. The bridge seems to depressurize, but no physical problems are suffered by either Jedi. Why do neither die, or in the least, pass out due to the lack of oxygen?
The implication seems to be that air was being evacuated not only from the bridge but also the surrounding corridors and body of the ship. Note that the screenplay makes it explicitly clear that the Jedi (and the Chancellor) were still being buffeted by escaping air even as the doors closed, indicating that there was air being drawn in from elsewhere. This air would have then filled the bridge once the blast door has closed.
OBI-WAN tries to get at GENERAL GRIEVOUS. ANAKIN runs at the General from the opposing side. GENERAL GRIEVOUS turns and throws his electrified staff at the window. It breaks, causing chaos as everything that is not nailed down is sucked into space. GENERAL GRIEVOUS is the first one sucked out into space. He fires a cable from his arm that attaches to the ship. He swings in and lands firmly on the side of the ship.
OBI-WAN, ANAKIN, and PALPATINE hold on for dear life. A blast shield closes around where the window used to be.
This is backed up by the film's junior novelisation which notes that there's still a "storm of air" even as the doors close.
As he was swept away from the ship, he pointed at the hull and triggered the built-in cable in his arm. The anchor struck solidly, attaching to the hull. He let the cable pay out until the automatic blast shield snapped shut over the broken viewport, cutting off the storm of air rushing out of the ship. Then he swung himself onto the ship’s hull, his clawed metal feet digging in.
I think we can be reasonably sure that the Nemoidian ship, despite being presently crewed by droids, has ample atmosphere to cope with a windshield failure.
There is no detailed canon explanation, but Matthew Stover's (Disney canon) ROTS novelization implies that the air wasn't fully sucked out of that space.
On the bridge, a blast shield had closed over the destroyed transparisteel window, and every last surviving combat-model droid had been cut to pieces even before the atmosphere had had a chance to stabilize.
Moreover, an extra confirmation is because Grievous, who knew exactly what was happening, (1) decided to escape instead of waiting for Jedi to asphyxiate and (2) he explicitly mentally noted that he expected them and Palpatine to burn up on re-entry, not asphyxiate now:
He cast off the cable. His hands and feet of magnetized duranium let him scramble along the hull without difficulty, the light-spidered curve of Coruscant’s night-side whirling around him. He clambered over to the external locks of the bridge escape pods and punched in a command code. Looking back over his shoulder, he experienced a certain chilly satisfaction as he watched empty escape pods blast free of the Hand’s bridge and streak away.
All of them.
Well: all but one.
No trick of the Force would spring Kenobi and Skywalker out of this one. It was a shame he didn’t have a spy probe handy to leave on the bridge; he would have enjoyed watching the Republic’s greatest heroes burn.
