TL:DR - don't faff about with real plaster for such a small repair… plaster is great for large open surfaces, but rubbish for patching tiny cracks. It's neither sticky enough nor penetrative enough.
That doesn't look like the world's oldest plaster, but you don't really know whether that's just an isolated patch or the rest is just held up by the the top skim.
Unless you're planning on re-plastering the whole lot & so long as you don't expect it to hold up that light fitting [which you should find a joist to fasten to] then sure, PVA is a pretty good semi-permanent strengthener.
It's much harder to deal with it on ceilings compared to walls, so you need to prepare for some spillage & dripping until you can get a good penetration & time to set, but it will work to an extent. Use too much, too thin & catch the drips/spills, for maximum penetration. Leave each 'coat' to dry before going again, rather than trying to get it all to stick solid in one go.
Once done, give the top a light skim with 'Polyfilla' or similar. DIY plaster patching in a tube, with flexible plasticisers to make it very sticky & resilient. Soaked-in PVA & filler will both take paint well.
I live in an old Victorian pile with plaster which has aged to little more than hairy dust these days. There's no way I'm going to re-plaster the lot, so I've injected much of it with thinned PVA & can report that the result is considerably more stable, sufficient that I can now drill through it into the brickwork behind without it feeling like I'm drilling dry sand.
Injection for small areas can be done with a blunt syringe - these days you can get them from shops that sell 'vape' accoutrements as well as chemist/pharmacists. The size of the syringe & blunt needle will dictate how thin the PVA needs to be.
One last note.
If it's just a bit cracked rather than completely decaying to dust, use 'No More Nails' or similar high-grip/rapid dry adhesive instead. You can fill like plaster, just injecting the cracks until you can tell you've pushed adhesive right through. Result is stronger than plaster or PVA/plaster, but it doesn't penetrate. Again, skim with Polyfilla afterwards.