You can't just buy some WiFi dongle and expect it to work. You will need some sort of USB device server, whether standalone or built into a router. Also, getting Time Machine to work is a bit more complicated.
If you have an AirPort Extreme or Time Capsule, it includes a USB port in the back that will allow you to share USB drives. The Express will not however, as the USB port it has is only for printer sharing. Other routers may have USB device servers that will allow you to share the USB drive, but be sure to get one that is compatible with Macs.
You can also purchase standalone wired or WiFi USB device servers.
This offering by Edimax has Mac support, but may require a Windows PC (or Windows on your Mac) for initial setup if my experience with their products is any indication. After setup however, you will be able to access it via your Air. That said, it will require you to reformat your HDD as FAT16, FAT32, NTFS, EXT2, or EXT3 (no support for HFS+) and it seems only capable of FTP or SMB (AFP is required to use Time Machine without a hack). A Google or MRoogle search may bring up other products that will work, but from the preliminary searches I've done it seems Mac support is sorely lacking.
If none of those solutions are acceptable, and if you have another Mac, you can connect the USB HDD to that and share it on your network. So long as that Mac remains on and on your network, you will be able to access that drive via your Air and even use it for TM I believe.