Get a 16 gigabyte usb thumb drive/memory stick, those cost only about $10 now. That's the current sweet spot for price/size.
just find one that has free shipping and you're good to go. 8 gigabyte is less and costs under 10.
If the public computer allows you to plug those in, save the downloaded files to the thumbdrive, and be happy, and you'll have your music.
Music files have little to do with wmp, unless it's specifically apple protected types, like from itunes. Any and all music players will play mp3s, most will play mp4s. Most will play wma type files, but I urge you to avoid those, as well as the apple itunes formatted stuff, usually called aac, which is just a copy protected mp4 I believe. Avoid anything with copy protection, it's a pain and a headache.
And you can always download and run foobar2000 if you get your own pc, that plays everything, that's windows program.