Welcome to the forums!
Does your phone have a Pattern/PIN/Password lockscreen?
If not, you should be able to connect it to your computer and use the Kies software to backup almost everything.
Your contacts should already be synced with Google, and your email lives in the cloud anyway.
If you're getting the same model as a replacement you could temporarily swap the screen over to unlock and backup the old one, then reconnect the screen to the new one.
Or you could try something like
http://www.amazon.com/dp/B009QW3TBK...ag=hawk-future-20&ascsubtag=UUacUvbUpU3567379 which should mirror the screen on your TV, and you could use a USB keyboard & mouse to navigate.
Sent from my rooted, debloated, deodexed Sinclair ZX-80 running CM -0.001 using Tapatalk