You need to sync your old iPad to your computer and add all the apps and stuff you want on it and then sync your new iPad to the computer and put everything on to the new iPad
Now for your problem I'm not sure if mystery manor runs of game center or open feint if it does you need to log into game center or open feint and all your progress and friends will be their or if the game has it's own log in system and if it doesn't..sorry your out of luck