Time Machine without Airport?
Time machine requires a specially formatted hard drive - it doesn't have to be an Apple drive and indeed can be anything you can connect to the machine, but it must be HFS+ and as such cannot be shared by Windows (FAT/NTFS). 

My choice in this situation (I too have a W7 machine too) was to purchase a relatively cheap Western Digital Portable HDD from Amazon, let my Mac format that and use it exclusively for backups. I couldn't find anything to share them between machines at all. 

If you're only looking to backup files, however, Dropbox may suit you well. It's free plan provides you with a 2gb folder that syncs between your computers (and the cloud). 

I'm interested in what you said about an Airport router - your Macbook should be able to connect to any Wifi A/B/G network (so pretty much anything, for instance the free router your internet supplier provides).