I believe if you go to your account online, you can get to a section that lists your purchase history. It lists all apps you downloaded, even the ones that are free. The nice thing is you can start a re-install right from there, and you won't have to pay again for apps. So without touching the phone you can re-install all apps.
When I upgraded to a Lumia 920, that is how I did it. At one point the phone showed it was downloading/installing 57 apps at the same time (well, queued at least). All apps installed without issues.
Be sure to be on WiFi before doing that though.