I also serious doubt that it will work. Apple always says connect to wall power to do upgrades. Should a battery lose charge or fail during an upgrade, recovery can be a hot and uncertain mess.
Loss of charge will be more likely if you try to upgrade wirelessly. However, using wired Ethernet (preferred) requires an adapter for that model.
Did the eBay ad state there was no charger? If not, you need to have a serious conversation with the seller. As far as I can tell, Apple no longer sells chargers for that model, which uses a 45W MagSafe 2 Power Adapter
And see this, which applies to Mac computers as well as iDevices:
The All Too Common SAD Reality of Buying … - Apple Community