This is a very good question you have posted. At this time the Xbox 360 will not stream the Slingbox. Withthe browser Microsoft offers, you can not install the plug in to watch the Slingbox content. This does not matter if you are trying to watch a 350 or a 500. At this time there are 3 ways to watch the Slingbox and I will post them below.
For your setup I think that the connected devices would be the best for you and would allow you to watch that Slingbox on a TV without tying up your computer or your Mobile device.
Hope This Helps
Sling Moderation Team