The Xbox 360 does not support standard USB Wi-Fi adapters. It uses a proprietary wireless protocol requiring the official Xbox 360 Wireless Network Adapter (black "shark fin" or white "N" model).
Connect Xbox 360 via Ethernet to a laptop/PC → Bridge laptop's Wi-Fi + Ethernet adapters in Windows Network Connections → Share internet. Adds latency, but works without official adapter.
Verdict: Use Ethernet. It's cheaper, faster, lower latency, and never drops. Save the adapter money for games.