You need a 350Mhz receiver, and luckily you have one in the box!
You need to find the pin on the receiver that outputs the digital stream of 1's and 0's, use as scope hopefully a dso and grab figure out how the data is formatted and the speed. What buttons send what data, ect...
Then you need to buy a 350Mhz transmitter and send the same 1's and 0's to the receiver, at the right bit rate. Sparkfun should sell one.
You should be able to find a lot of information on this by goggling "reverse engineer a garage door opener"
I did this a few months back, it's not too difficult, the biggest problem will be getting the speed right, and the distance that you want. distance at 350Mhz is a problem