My hard drive crashed, tons of songs on my iphone 5 and can't add new ones from PC without completely erasing it, what can I do?

My hard drive recently crashed and I cannot recover it. I have a new hard drive and have re-installed iTunes, I have authorised this PC already and I used to have my iphone with "manually manage music", now I can't add nor set "manually manage music" to my iphone without having to erase it, I don't want to have to erase my iphone and have to re-download everything. What can I do? I have very slow internet and it will take weeks if not months to re-download all of my stuff/apps/songs again.

Will backing it up, erasing it, then restoring it again fix this? Will I still have to download my songs/apps again? Will it even let me back it up since I have to erase before syncing with this new pc?