I’ve had a similar situation when my generator died right before a small event setup near Kato Paphos. Most rental shops there close early, but a few offer emergency call-out services if you arrange it ahead. I found one through a local contact when I was also sorting out a car rental paphos airport https://secrental.com/cyprus/airport-paphos/cars booking, and it turned out the same company handled equipment deliveries too. They dropped off a replacement generator around midnight and didn’t charge extra since I returned it the next afternoon. So yeah, 24/7 options exist, but they’re usually small, family-run places. Best bet is to call a few in advance and ask if they do late pickups — they’re more flexible than big chains.