(it's simultaneous, so you gain 0 net life, which triggers nothing, right?)
You still get the life gain/loss trigger. Even though the net result of the ability is 0, triggers can be "created" (pending priority) in the middle of abilities resolving.
603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
603.2a Because they aren’t cast or activated, triggered abilities can trigger even when it isn’t legal to cast spells and activate abilities. Effects that prevent abilities from being activated don’t affect them.
Ajani's Pridemate would trigger in this scenario. When Orzhova's ability resolves targeting yourself, you will lose life, then gain it back. The key there is that you do gain it back, meaning you did gain life - and that's all that Ajani's Pridemate cares about.
In this situation I would take the card at face value. What it is telling you is that you can have any target player lose 1 life. Then it states you gain 1 life. Even if the player you chose is yourself, you lose one then gain one in that order. Any triggers that happen as a result of: first the life that was lost, then the life that was gained, will all still trigger regardless of the player that was chosen to lose 1 life.
I would add that I'm not certain of any reason why you would have yourself lose one life from this ability but I'm sure someone here will be able to think of a situation where the life loss would help you if triggers happened from it.