Yeah, I think 3rd party apps are the only way to go in this case. You can do a DL-play off your router if you don't have a flash card, but you'll either need a passthrough or flashed ROM to play non-signed apps.
I've seen several that use ssh, and the homebrew boards are always talking about new ways to do it.
Of course, you could just wait until the Opera browser cart is put out by nintendo. I think that would be a good solution to your problem.
http://www.opera.com/products/devices/nintendo/