If this is a PowerPC (as opposed to Intel) Mini, then you need to use the SlingPlayer for Mac app, as the web browser plugin won't work on older PowerPC systems. I encountered the same issue when using an old iBook connected to a big-screen LCD TV (and had to use the SlingPlayer for Mac application instead of a browser and the Sling.com plugin).
Also, keep in mind that since Sling has killed development of SlingPlayer for Mac (never did get that HD streaming, dangit!), you will get a fake "update available" notice when you launch the player. You can see this notice and get the actual Mac app here (the link to download the SlingPlayer for Mac is on the far right, via the nearly-hidden "SlingPlayer for Mac Desktop Software Download" link):
Hopefully they'll keep this player available, even if stagnant, since the web-based stuff is often such a hassle...