The water heater in the apartment above me burst and, naturally, flooded. Also naturally, it leaked into my apartment in various places. One of those places was the kitchen ceiling light. After the leaking stopped, I removed the glass shade (which was full of water) and gave the fixture 24 hours to dry.

Is it safe to turn on now or should my landlord do something about it?

Many fixtures have fiberglass insulation above the canopy. If this got wet, it will stay wet for a while, and cause problems down the road.

The fixture should be taken down, inspected for water inside and dry or remove and replace any wet components. Check all splices for water and corrosion.

If it is a simple, cheap fixture, better to just replace it.

Agreed but your LL should be doing this, not you. You should not repair or replace any fixture or appliance in a rental unit without the LL's permission and financial consideration.

Don't want to start a post, anyone see a bulb fill or half fill with water from a leak above?
Seen it about 5 times under carports?

I've also seen a light bulb half filled with water. It was on the scaffold of a dock crane.

We used to have a joke about "4-phase." That was when you had three wires and water coming out of the same pipe.