Old plaster ceilings can absolutely hold a ceiling fan, but you need to do some homework first because plaster is more brittle and unpredictable than drywall. The main issue isn't the plaster itself - it's what's behind it and how solid the connection is to the framing. Most ceiling fans weigh somewhere between 30 and 50 pounds, which isn't a ton, but concentrated in one spot through a small mounting bracket means you need solid support.
Start by checking what's actually up there. If you have attic access, get up there and look at the joist structure directly above where you want the fan. You're looking for solid wood joists that you can mount to. Tap on your ceiling from below in different spots - solid spots sound different from hollow spots. If you're hearing mostly hollow sounds or the plaster feels soft or crumbly when you tap, that's a red flag. You might also notice cracks radiating from light fixtures or other ceiling-mounted stuff, which suggests the plaster or substrate has some issues. If the plaster is in rough shape, you may need reinforcement before mounting anything heavy.
For reinforcement, the standard approach is to install a fan-rated ceiling box that's specifically designed to bridge between joists and distribute weight. These boxes have cross-bracing that runs between two joists, which is way more secure than just relying on plaster. You'll likely need to cut a hole in the ceiling, install the box, then patch the plaster around it. If you're not comfortable doing that yourself, it's definitely worth having someone take a quick look at the structure before you mount anything. An electrician can handle the wiring, but they might not catch structural issues, so a quick attic inspection on your own beforehand saves headaches later.