I don't think it is possible, as I have looked and continue to look 4 solution.
I have a HDMI TV, with an mk808b google mini TV stick running 4.2.2 I purchased from Amazon for $60 at the time
http://www.amazon.com/gp/aw/d/B00AMX47AY
and a wireless keyboard, also comes with a USB wireless connector for other devices (Bluetooth also available)
http://www.amazon.com/gp/aw/d/B003UE52ME/ref=pd_aw_sims_6?pi=SL500_SS115, or you can use the remote app on your rush to control the device. Anyways, it displays android as a tablet on my TV, I installed a custom ROM by Bob called "finless"
http://www.freaktab.com/showthread.php?4966-NEW-MK808-B-(original)-JB-4-2-2-Official-Finless-ROM-2-1 witch has built in options to send things from my galaxy rush "stock ROM" to my TV like Chromecast. I also have apps like cheapcast, ehome media center, xbmc, and a couple of other apps to make the experience better. With this there is no need to connect your rush to the tv really, it's cheap, gets what i need done, and the family loves it more then me! lol. But to answer your question, I don't think it is possible as of now with this device, but I could be wrong.
The rush has a built in feature in the stock samung video and music player that allows it to connect to another device and stream it's content when on the same WiFi network.