From the old West End Games D6 Roleplaying Game (Second Edition) supplement Death Star Technical Companion, Chapter Two: Technical Specifications, p. 16, right column:
Gravity within the battle station was handled by omni-directional gravity boosters built into the decks, walls and ceilings. These gravity boosters changed orientation as easily as flipping a switch, and they were designed to allow the gravity orientation to be altered from sector to sector, or even from corridor to corridor. While hangar bays imposed gravity perpendicular to the Death Star's core, adjoining corridors shifted the gravity orientation to coreward. In a situation where the gravity orientation changed from one section to the next, there were numerous warning signs. However, most gravity orientation transitions were accomplished by turbolifts, which employ gravity compensators oriented to the lift's floor. While the lift was in transit, it would rotate to match the orientation of the destination deck, while the compensators would keep the occupants perfectly comfortable and completely unaware that the gravity orientation had changed at all.
To go into the specifics of the question:
Gravity does -- for the most part -- work as on Earth (or any other celestial body), and you are upside down in the lower hemisphere, or rather, Space is always above you (which is a good thing while, for instance, manning a turbolaser battery).
- The gravtiy changes are carefully marked wherever they occur (near a
hangar, probably also near the core or the superlaser maintenance
ducts - otherwise those poor guys would have to climb a 120km (or
160km in DS-2) long ladder). Elevators (i.e. turbolifts) have their
own gravity, so the capsule rotates but noone inside feels it, you
just get on and off and maybe in between the gravity has changed
orientation a few times, but you wouldn't even know.
Only within a specific area do the people experience the same gravity.