If you want to keep a Windows PC running 24/7, the free AcuRite My Backyard Weather (MBW) software and web site can be configured to send your station's USB port data to Weather Underground from the MBW web site. If you don't want to run your PC 24/7, the bridge (which has been unavailable for months) will upload to MBW without a PC, which you can then use to feed WU. As it sits today, I don't believe that the bridge offers any third-party software support.
The only third-party software support for the USB port is via Weather Display,
http://www.weather-display.com/. Both it and the AcuRite MBW software use the same external Windows-based USB downloader program to yank data from the 1035 console from what I've read, but I have not personally tried WD with my own 1035 yet. Everything is Windows only right now.
I cannot comment on how well the feed from MBW to WU works, since I am doing the public a service by *not* publishing my data to WU. My outdoor sensor is mounted near several obstructions, and there's enough ASOS sites within 5 miles of my house where the data isn't really needed.....