|Before updating Outlook you should have "exported" all your emails & contacts as a 'PST' file from the old Outlook, then "imported" it into the new Outlook after updating it.|
I'm surprised you didn't think to keep a backup of your emails because it's so easy to do with Outlook. It's something you should always do in case the hard drive fails.
You might get lucky and find a file called Outlook.pst in your User Account folder here: C:\Users\<My account name>\AppData\Local\Microsoft\Outlook
You will need to have "Show hidden files & folders" enabled to be able to see those folders.
Import that file into Outlook 2010 using File > Import & Export