Apps - no as the old apps do not work on the Z10.
BlackBerry Link has replaced Desktop Manager. It has the ability - which has been difficult for some but appears to be getting better - to move your contacts, notes, calendar and tasks.
More and more apps are being updated daily to work with BB10, so check BlackBerry World for the apps you want to know about. You would need to download the BB10 app version and if they are paid apps, you will have to pay again.