The fan indeed works properly, and after like 30 seconds spins at full speed... I can feel the air coming out of the cooling vent pretty strong, and after a minute and a half it gets pretty warm.
Product number is KU247ES#AKN
It has WinXP (Service Pack level unknown), same problem even w/o the Hard Drive.
The WiFi card is a Broadcom BCM4311KFBG.
I got this laptop to try to fix it (the owner described it was shutting down).
I opened it and I tried touching the cooler where the processor is and it was pretty hot.
I removed the cooler and saw that the thermal grease was solid, not liquid anymore, so I removed the old one and put a new one.
I reassembled the laptop but it still shuts down, the cooling vents are dust-free, the cooler is properly attached (screws tightened).